国人过于高看诺贝尔奖了

Originally published at 狗和留美者不得入内. You can comment here or there.

 

今天在已发表至少一个月而得到不少赞同的此文章改了个错别字,完了却被删除了,我把内容复制如下。


今天的诺贝尔奖的科研工作意义显然远不如前,因为基础科学已经遇到瓶颈了。今天的科研也超级专及窄了,与二十世纪上半期所出的更基础通用简单的科研工作截然不同。1920年代有俩中国实验物理学家吴有训赵忠尧在美国读博士时就做出了载入史册的至少接近诺贝尔奖级别的关于散射的科研工作。那个时候的划时代散射实验几个人在实验室就可以做,与今天的需要投入上亿的对撞机也截然不同。杨振宁李政道发现的宇称不守恒在理论科学角度是划时代的不得了的,虽得了诺贝尔奖,但也没啥实际的影响,与二十世纪初期的更基础的好多理论物理很不同。中国接受源于西方的近代科学和工业比较完也注定了中国没跟上二十世纪前半期的科技的突飞猛进。就1950年,中国需要解决的是实际问题,被瑞典二留科学家凭的基于炸药资本家的遗产的基础科学奖对中国意义也没多么大。

那么多不了解科研怎么回事儿的国人把诺贝尔奖获得者似乎视为万能的神,这是搞笑的。发现DNA机构的James Watson晚年由于政治不正确的关于基因的言论被公开排斥了,他的有精神分裂症而无法自理的儿子估计也给他添加的不少经济负担,据说晚年的他缺钱而把他的诺贝尔奖以一百万多卖给了某富翁。我也晓得德国人Wolfgang Ketterle,这人为了实验实现接近于零温度的物质态获得了零几年的诺贝尔奖,获奖时他离婚,感觉法庭给他的条件比较刻薄,要求他出住了抑郁症试图自杀而住了精神病院的老婆的医疗费并给三个孩子付大学学费,毕竟他作为麻省理工学院教授二十几万的年收入远低于华尔街quant的,他也基本不可能发什么财,他的科研也无实用价值。他与俩其他人分享诺贝尔奖税后得到的钱估计不到20万刀,美国的权贵根本不会了他,普通老百姓也基本不会在乎,跟好多中国人对诺贝尔奖的焦虑或关注很不同。就是在中国,杨振宁姚期智丘成桐虽是大科学家但跟中国的政治主流也基本没啥关系。

就不用说今天随着西方衰落的趋势,西方人包括西方名校教授的评价越来越无关了。中国的权贵甚至普通人完全可以以fuck you的态度对待美国名校教授的,因为他们跟中国也没啥关系,在中国也没有中国公民的权利,社会关系基本限于日益衰落的西方国家。我鼓励国人多关注一些自己的事情,而非围绕着诺贝尔奖而崇洋媚外。西方人得到的诺贝尔奖远远更多也并不说明西方人更聪明,更多因为积累的因素。这个跟教育体质也没啥关系,你去哈佛读本科跟哈佛的诺贝尔奖得主教授也接近100%的概率不会有啥关系的,人家忙得很,也有自己的生活和圈子,那顾得上了本科生,就他们的博士生大多也最多每周见一次。行外,他们若不懂也完全没资格讨论,就斯坦福理论物理教授祁晓亮,哈佛大学理论物理教授尹希之类非常政治幼稚地表示反中国体制内的观点,跟他们的理论物理研究做得多么好也没啥关系。

How weak convergence and weak topology arises naturally from infinite dimensional vector spaces

Originally published at 狗和留美者不得入内. You can comment here or there.

Let [公式] be a vector space over [公式] endowed with an inner product [公式] . Assume that it has an orthonormal basis [公式] . This means any [公式] can be expressed as

[公式] where [公式] for all [公式] . For each [公式] , we take any arbitrary sequence of reals [公式] that converges to [公式] . Let [公式] . We can also require that [公式] for all [公式] and that [公式] , which means uniform convergence across all [公式] . If we want that a cofinite number of the sequences at index [公式] are at least a certain distance away from the value converged to, we additionally define[公式] and require that for all [公式] , [公式] . Obviously, [公式] for all [公式] . Setting [公式] obviously satisfies these requirements.

We notice that [公式] iff the associated series converges, which of course does not happen all the time, which means that that [公式] via [公式] is not well defined. Thus, the two-norm on finite dimensional vector spaces induced by the inner product cannot be defined similarly on infinite dimensional vector spaces.

Proposition 1 There exists an infinite dimensional normed vector space that is a Banach space.

Proof: Keep in mind that for all [公式] , we must have [公式] one of the properties of norm. For simplicity we shall require that for any [公式] such that [公式] ,

[公式] which is of course compatible with triangle inequality. We shall require that [公式] is a Banach space which means that if [公式] is convergent, it is in [公式] . Moreover, we require that if [公式] converges, then [公式] . We now show that this is well-defined. We note that by [公式] ,

[公式]

Assume that the infinite series converges, by definition, for any [公式] ,

[公式] Then, that

[公式] implies that

[公式] Applying absolute homogeneity of the norm to [公式] gives us

[公式] If [公式] is convergent, then [公式] must of course be bounded based on our extension of norm to the infinite series. Assume it converges to [公式] . Then by [公式] , [公式] . Thus, we have

[公式]

which verifies absolute homogeneity. Verifying triangle inequality is straightforward and we leave this to the reader.

For [公式] , it is necessary that [公式] for an arbitrary real sequence [公式] . By the aforementioned properties, if we prescribe values for [公式] for all [公式] . If we require that for any [公式] , the associated coefficients satisfy [公式] , then values for [公式] such that [公式] suffices. It is easy to see that for any sequence of absolutely bounded real coefficients, there is some upper bound [公式] , which implies that the norm of the vector in [公式] associated with it is upper bounded by [公式] .

Finally, one easily verifies closure under scalar multiplication and addition of this vector space, wherein the vector coefficients are guaranteed to have bounded supremum.

We note that the bounded supremum requirement is the more essential part here. There is a in fact simple stupid way to construct a norm which is given by [公式] . In order for this to be a norm, we must of course always have [公式] . [公式]

Example 1 Let [公式] . Then, [公式] .

Example 2 We can let [公式] .

Example 3 We can define a sequence [公式] of vectors converging to [公式] via sequences of coefficients [公式], we indeed have [公式] with respect to this norm. More specifically, [公式] goes to [公式] as [公式] . Thus, sequence converges with respect to norm given in Example 2. It also does with respect to the norm given in Example 1.

Example 4 If we let [公式] for all [公式] , then we would need to restrict the vector space elements to correspond to sequences [公式] such that [公式] .

Example 5 We let [公式] . Using the norm given in Example 1, we have that for all [公式] , [公式] . The [公式] tells us that it also cannot converge to any value in other [公式] . Thus [公式] does not converge with respect to the Example 1 norm. It does converge to [公式] with respect to the Example 2 norm though.

Proposition 2 Any [公式] is uniquely defined by prescribing the values of [公式] over all [公式] . In order for [公式] to be well-defined, we must have [公式] .

Proof: Follows directly from definition of basis and linearity of [公式] . [公式]

Definition 1 We say that [公式] , or that [公式] converges to [公式] strongly with respect to some norm [公式] iff [公式] .

Definition 2 We say that [公式] , or that [公式] converges to [公式] weakly, iff for all [公式] , [公式] .

Proposition 3 The sequence in Example 5 converges weakly (in the Example 1 norm).

Proof: If [公式] , then by Proposition 2, we have [公式] . Let [公式] and [公式] , with of course [公式] monotonically from above.

Then, we have that [公式] . That this upper bound converges to [公式] from above completes our proof. [公式]

Definition 3 The coarsest topology on normed vector space [公式] with respect to which every [公式] is continuous, which we shall denote with [公式] , is the weak topology.

Proposition 4 [公式] iff every neighborhood of [公式] with respect to [公式] contains for some [公式] all [公式] such that [公式] .

Proof: Follows directly from the definition of [公式] and of neighborhood. [公式]

Proposition 5[公式] is a neighborhood basis of [公式] . If we let [公式] denote an arbitrary finite subset of [公式] , then the every set of the aforementioned collection can be denoted as [公式] .

Proof: Satisfaction of the continuity requirement means that for any open interval [公式] in [公式] , [公式] for any [公式] . We thus take [公式] to be a subbase of [公式] . (For the definition of subbase, see Definition 1 of [2].) Every open interval in [公式] has a midpoint which we take to be [公式] , and we take [公式] to be half the interval length. The collection of sets given in the proposition are all finite intersections of elements of [公式] . Every [公式] is the arbitrary union of finite intersections of elements of [公式] , by definition of subbase. Thus, either [公式] is the empty set, or it contains some [公式] . This completes our proof. [公式]

Definition 4 The strong topology of a normed vector space is the coarsest topology such that the norm is a continuous function.

The norm on the dual space [公式] is defined as in Definition 8 of [3]. This norm gives rise to a strong topology on the dual space as well.

Lemma 1 For any [公式] , [公式] is closed in the strong topology on [公式] .

Proof: [公式] is a closed set and the norm is a continuous [公式] function. Thus, [公式] is also closed. Proposition 7 of [4] tells us that the preimage of a closed set, when the function is continuous is also closed. [公式]

Proposition 6 For any [公式] , [公式] .

Proof: We note that if not, then [公式] would not be a well-defined norm on [公式] . By definition of [公式] , [公式] is continuous. By Proposition 2 of [3], continuous implies bounded. The final equality in Definition 8 of [3] then gives us the desired result. [公式]

Proposition 7 For any normal vector space [公式] , the weak topology is a subset of the strong topology and strong convergence implies weak convergence.

Proof: It suffices to prove that strong convergence implies weak convergence. Assume [公式] . Then, by Proposition 6, for any [公式] , [公式] , with [公式] , which completes our proof. [公式]

References

Using the Minkowski functional to prove separation of sets via a hyperplane

Originally published at 狗和留美者不得入内. You can comment here or there.

We shall use n.v.s to refer to normed vector space.

Definition 1 Let [公式] be a n.v.s. A set of the form [公式] is called a hyperplane.

Definition 2 We call a set of the form [公式] a half-space determined by [公式] . Replacing [公式] with [公式] gives the other half-space.

Proposition 1 Any half-space is a convex set.

Proof: Trivial. [公式]

Proposition 2 Let [公式] be a n.v.s. [公式] such that [公式] . For any [公式] , the hyperplane [公式] is closed iff [公式] is continuous.

Proof: [公式] open iff [公式] is closed. Suppose [公式] is continuous and that for [公式], [公式] . Take some open neighborhood [公式] of [公式] not containing [公式] . Then, [公式] is open and disjoint with [公式].

To prove that [公式] is continuous, it suffices to prove that [公式] is bounded, by Proposition 2 of [1]. To show that [公式] is bounded, showing that [公式] suffices, where [公式] is the unit ball centered at zero. To do so, it suffices to show that every [公式] such that [公式] has an open neighborhood of the form [公式] such that for any [公式] , [公式] , since if this holds, then for [公式],

[公式] Thus, [公式] .

Assume that [公式] is open. Then, every [公式] such that [公式] is contained by a [公式] . Suppose that [公式] satisfies [公式] . Then, there must exist some [公式] such that[公式] , a contradiction. This completes our proof. [公式]

Definition 3 Let [公式] be two subsets of [公式] and [公式] with [公式] . If [公式] is such that of its two half-planes, one contains [公式] and the other contains [公式] , then we say that [公式] separates [公式] and [公式] . We say that [公式] strictly separates [公式] and [公式] if there exists some [公式] such that

[公式] We now wish to prove that for any open convex set [公式] containing [公式] and any [公式] , there exists a hyperplane that separates [公式] and [公式] . Let [公式] denote the linear functional associated with this separation. We prescribe for [公式] some [公式] . If we can find some sublinear function [公式] that is strictly bounded above on [公式] by [公式] such that on the subspace [公式] satisfies [公式] , then we can apply the Hahn-Banach theorem to extend [公式] to all [公式] in order separate [公式] from [公式] by a hyperplane.

For this desired sublinear function [公式] , we try the following.

Definition 4 Let [公式] be a subset of n.v.s [公式] . Then the gauge or Minkowski functional with respect to [公式] is defined by [公式] .

Proposition 3 If [公式] contains an open ball centered at [公式] , the Minkowski functional [公式] satisfies the condition that for all [公式] , [公式] .

Proof: Trivial and left to the reader. [公式]

We want that on [公式] , [公式], which is satisfied when we let [公式] . We let [公式] . That [公式] is open means there is some open ball centered at the origin [公式] . For any [公式] , [公式] obviously holds. This is a simple stupid way to uniformly upper bound on [公式] .

Proposition 4 For any subset [公式] , [公式] is bounded above by [公式] for [公式] . If [公式] is open, then for all [公式] , [公式] .

Proof: Trivially proven. [公式]

Proposition 5 If [公式] is an open convex set containing [公式] , then [公式] .

Proof: Since we can scale down arbitrarily by Proposition 3, it suffices to prove this triangle inequality on an open ball centered at [公式] that is contained by [公式] . Moreover it suffices to prove that for any [公式] ,

[公式] is satisfied. We notice that

[公式]

with[公式]

Let [公式] . Obviously [公式] . With this in mind, that [公式] is convex implies that for any [公式] for any [公式] , [公式] . We use [公式] to set the value of [公式] noticing that we can multiply by the denominator to derive the desired inequality. We calculate

[公式] We notice that if we replace [公式] with [公式] , we would obtain the desired inequality. Because [公式] is convex [公式] must be connected. Assuming that [公式] , we have [公式] . The connectedness hypothesis implies in fact that all [公式] such that [公式] must be in [公式] . We have that

[公式]

which then implies [公式] for [公式] . Thus, we are able to make the aforementioned replacement, which then completes our proof. [公式]

Proposition 6 If [公式] is an open convex set containing [公式] , then its associated Minkowski functional [公式] is a subadditive function.

Proof: The two requirement of subadditive were proven in [公式] and [公式] respectively. [公式]

Theorem 1 Let [公式] be an open convex subset of a n.v.s [公式] . For arbitrary [公式] , there exists [公式] such that [公式] for all [公式] . In particular, the hyperplane [公式] separates [公式] and [公式] .

Proof: After a translation, we may always assume that [公式] . The Minkowski functional [公式] by Proposition 6 is a subadditive function. We define [公式] on [公式] to be linear. We can extend [公式] to [公式] on all of [公式] by the Hahn-Banach theorem (see [2] for details on it). Since by Proposition 4, for any [公式] , [公式] , we have for all [公式] , [公式] . Thus, setting [公式] gives us what we want to prove. [公式]

In [3], propositions of more general separation of sets by hyperplanes were proven. I shall not write them up here because I believe the foundational ideas behind their proofs have already been explained in the propositions above. I had learned of the Minkowski functional this week but initially did not feel like I grasped the motivation behind it. Such difficulty was resolved after reading 1.2 in [3], the title of which is “The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets”. Interestingly, I read about the Minkowski functional on Wikipedia before writing up [2], in the process of which I gained non-trivial understanding of the Hahn-Banach Theorem, which I had learned in 2017 or 2018 but forgotten in 2021 due to more or less superficial understanding. Certainly, the Hahn-Banach theorem can come across as very formal and abstract at first encounter, and it might not be immediate why it’s so significant. However, the geometric interpretation of it via separation of convex sets gives it more concrete meaning.

References

On the Baire category theorem and the open mapping theorem

Originally published at 狗和留美者不得入内. You can comment here or there.

Theorem 1 (Baire Category Theorem) Let [公式] be a complete metric space.

  1. The intersection of a countable collection of open dense sets is dense.
  2. [公式] is not a countable union of nowhere dense sets.

Proof: Let [公式] be our intersection of a (countable) sequence of open dense sets. It suffices to prove that [公式] does not contain any nonempty open sets, or equivalently, that any open ball [公式] intersects with [公式] . By closure under finite intersection, we have that for all [公式] , [公式] is open. That [公式] is an open dense set implies that [公式] is open, which means it contains some closed ball [公式] of radius [公式] . With [公式] in mind, we also have that [公式] for some closed ball [公式] of radius less than [公式] . In this fashion we construct a sequence of closed balls [公式] for which [公式] , which implies that [公式] . Cantor’s intersection theorem (Theorem 1 of [2]) tells us that [公式] is non-empty, which means [公式] is non-empty, which completes our proof.

A set is nowhere dense iff its complement is dense in [公式] . Suppose by contradiction that [公式] is a collection of nowhere dense sets that such that [公式] . The application of de Morgan’s law tells us that [公式] , which the intersection of countably many dense open sets. This is impossible since [公式] is obviously not dense in [公式] . [公式]

Definition 1 If [公式] and [公式] are topological spaces, then a map [公式] is called open if [公式] is open in [公式] whenever [公式] is open in [公式] .

Proposition 1 For metric spaces [公式] and [公式] , [公式] is open iff for all any ball [公式] centered at [公式] , [公式] contains a ball centered at [公式] .

Proof: Suppose by contradiction that [公式] is not open yet any ball [公式] centered at [公式] is such that [公式] contains a ball centered at [公式] . In this case, there exists a ball [公式] centered at [公式] such that [公式] contains an element on its boundary, which we shall denote with [公式] . Every point [公式] has a open ball neighborhood [公式] contained in [公式]. By our hypothesis, [公式] must contain an open ball centered at [公式] that is also contained in [公式] , which is impossible since [公式] .

For the other direction, [公式] open implies that for any ball [公式] centered at [公式] , [公式] is open, which means it must contained a balled centered at [公式] . [公式]

Proposition 2 If [公式] and [公式] are normed vector spaces and [公式] is linear, then [公式] is open iff [公式] contains a ball centered at [公式] when [公式] is a ball of radius [公式] about [公式] .

Proof: A normed vector space is a metric space. Thus by Proposition 1, [公式] is open iff for all any ball [公式] over radius [公式] centered at [公式] , [公式] contains a ball centered at [公式] . We can openly and bijectively map [公式] to [公式] via [公式] . It is easy to see that

[公式]

contains an open balled centered at [公式] iff [公式] is open. [公式]

Lemma 1 In a normed vector space [公式] , for any [公式] , [公式] .

Proof: Trivial from triangle inequality. [公式]

Lemma 2 Let [公式] be normed vector spaces. Let [公式] be a surjective linear map such that for all [公式] , [公式] contains a neighborhood of [公式] , or equivalently that every ball [公式] is contained by [公式] for some [公式] . If [公式] converges to [公式] and [公式] converges to [公式] , then [公式] .

Proof: By the argument in the proof of Proposition 2, we can replace in the hypothesis neighborhood of [公式] with neighborhood of [公式] and closed ball of radius [公式] centered at [公式] with the one centered at [公式] . Suppose by contradiction that [公式] . Because every neighborhood of [公式] has infinitely many points [公式] such that [公式] is arbitrarily close to [公式] , there exists some open ball [公式] centered at [公式] such that [公式] does not contain any neighborhood of [公式] . This open ball necessarily contains for any [公式] , [公式] . Thus, the hypothesis of the Lemma we are trying to prove has been violated. [公式]

Theorem 2 (Open mapping theorem) A surjective linear map [公式] between two Banach spaces is necessarily an open map.

Proof: Let [公式] denote the ball of radius [公式] centered at [公式] . By Proposition 2, it suffices to prove that [公式] for some [公式] , or equivalently, that for some [公式] , any [公式] such that [公式] implies that some [公式] satisfies [公式] .

That [公式] is surjective means that [公式] . By the Baire Category Theorem, some [公式] is not nowhere dense and thus its closure must contain an open ball. This tells us that [公式] must contain an open ball. Every non-zero [公式] corresponds to a one-dimensional subspace. The restriction of [公式] to that subspace is of course continuous. Thus, [公式] . For any [公式] , there is a neighborhood [公式] of [公式] such that [公式] . Thus, [公式] can not contain any such [公式] , which means that [公式] . This means that for some [公式] , [公式] .

We now attempt to show a slightly weaker statement than desired, namely that for some [公式] , [公式] . Any [公式] satisfies [公式] . That [公式] implies that [公式] has a larger diameter than [公式] . We already know that [公式] . We also observe that [公式] is equivalent to [公式] by the linearity of [公式] .

We want to upper bound a ball centered at [公式] . To do so, we note that [公式] is equivalent to

[公式]

We also note that [公式] , which means by linearity of [公式] , [公式] . Thus, that [公式] implies that [公式] . This tells us that

[公式]

Combining [公式] and [公式] gives us

[公式]

which equates to

[公式]

which tells us that [公式] suffices.

With Theorem 1 of [3] in mind, we try to construct a sequence [公式] such that [公式] for all [公式] and strict inequality for at least one [公式] , which means that with [公式] , [公式] converges to some [公式] such that [公式] . This is because if for any [公式] , we can construct such a sequence such that [公式] , our proof is complete. By linearity of [公式] , [公式] gives us for all [公式] ,

[公式]

We notice that for any [公式] , there exists [公式] such that [公式] . Then, by [公式] , there exists [公式] such that [公式] . If [公式],

Let [公式] , the distance between [公式] and [公式] . If [公式] , then [公式] for [公式] such that [公式] is in [公式] . In this case, we can perturb [公式] and [公式] accordingly so that [公式] . Inductively, we derive that for any [公式] for all [公式] , there exists [公式] such that

[公式]

Moreover, for each [公式] , the finite set of [公式] s associated with it contains the finite set of [公式] s associated with all [公式] such that [公式] . In this way, we have a sequence [公式] and a sequence of partial sums [公式] that converges to some [公式] . Via [公式] , the sequence [公式] , which converges to [公式] is induced. By Lemma 2, we have that [公式] . This completes our proof. [公式]

References

Some “gmachine1728” on the internet reminded me of the j-variant

Originally published at 狗和留美者不得入内. You can comment here or there.

https://en.gravatar.com/gmachine1728

1729 = 10^3+7^3 = 12^3+1^3 is a number associated with Ramanujan. 1728 is also uniquely mathematical, in that it appears in the j-variant of the theory of complex multiplication. Though I have no actual detailed knowledge on the aforementioned mathematical topic as of June 2021, I shall quote the following.

Strong convergence in artistic tastes amongst people at the extreme right tail. Another point of convergence is the theory of complex multiplication in math, involving the j-invariant, moduli spaces, and abelian extensions of imaginary quadratic number fields.

For example, the great mathematician Barry Mazur writing

The elliptic modular function is loved by the analysts, arithmeticians and algebraic geometers who study elliptic curves since the isomorphism class of the elliptic curve formed by the lattice generated by the complex numbers 1 and z is completely determined by j(z), usually referred to as the j-invariant of the elliptic curve. It is the showcase example of a modulus in algebraic geometry, i.e., a continuous parameter that classifies a continuously varying array of distinct isomorphism classes of mathematical objects.

AND it was loved by Leopold Kronecker who hitched the aspirations of his youth (his Jugendtraum) on the ability of the elliptic modular function to help generate, in a magically ordered way, all algebraic numbers that are relatively abelian over quadratic imaginary number fields.

And David Hilbert saying

The theory of complex multiplication is not only the most beautiful part of mathematics but also of the whole of science.

And the great mathematician David Mumford writing:

Especially, I became obsessed with a kind of passion flower in this garden, the moduli spaces of Riemann. I was always trying to find new angles from which I could see them better.

I wonder who this gmachine1728 is, and in case he sees this, he is welcome to use this contact page https://gmachine1729.wpcomstaging.com/%e8%81%94%e7%b3%bb-%d0%ba%d0%be%d0%bd%d1%82%d0%b0%d0%ba%d1%82-contact/.

Or email me at gmachine1729 at foxmail.com. I have had some interesting, high-quality people find me through this blog and contact me, and I also became good friends with some of them, chatting with them a fair bit on the internet. So yes, readers don’t be shy. Feel free to contact me if you have something worthwhile to talk about or something you want to ask. I like to think of myself as an easily approachable person. I will certainly respect your privacy and won’t leak your email information to anyone else without your permission.

How to interpret the Hahn-Banach theorem

Originally published at 狗和留美者不得入内. You can comment here or there.

The motivation behind the Hahn-Banach theorem can come across to a functional analysis newbie as somewhat elusive. I shall here try to explain this to the extent that I understand it.

Suppose that a seminorm on a vector space [公式] is such that [公式] iff [公式] , for some subspace [公式] of [公式] . Let [公式] be the dual space of [公式] . If[公式] is bounded with respect to this seminorm, then [公式] . We wish to induce via this seminorm a norm on the quotient space [公式] . Since a norm induces a metric (and a seminorm induces a pseudometric), it is natural then to define the norm on [公式] to be the distance corresponding to the seminorm between [公式] and [公式] , which is formally [公式] . It is easy to verify that this is well defined and a norm.

Similarly, [公式] induces an element [公式] of [公式] . Since [公式] is arbitrary, [公式] is also an arbitrary functional on its domain. We wish to show that given the constraint that for some subspace [公式] of [公式], [公式] , we can for any [公式] , define functional [公式] on [公式] such that [公式] .

We are interested in extending a functional defined on a subspace to the full space with agreement of values on the subspace and a certain degree of boundnesses, more specifically an upper bound by the norm. Since this is trivially obtained by simply mapping to [公式] the elements outside the subspace, we are interested in an extension that is as non-zero or as large in absolute value as possible. In attempt to achieve this, we can try extending the functional with the requirement that its value on any input [公式] , which can be negative, is bounded above by the value of the application of another function on [公式] that has the reals, including negative ones, as its codomain. With this along with the properties of norms and seminorms in mind, we define the following.

Definition 1 Let [公式] be a real vector space. A sublinear functional on [公式] is a map [公式] such that for all [公式] and [公式] ,

  1. Triangle inequality: [公式] .
  2. Non-negative homogeneity: [公式] .

We immediately notice that the constraints defining a sublinear functional are a subset of the constraints defining a seminorm or norm, which means that any seminorm or norm is necessarily a sublinear functional, which means that any proposition that holds for an arbitrary sublinear functional also holds for an arbitrary seminorm or norm.

Lemma 1 We extend a linear functional [公式] , where [公式] is a subspace of [公式] , defined such that for all [公式] , [公式] for an arbitrary sublinear functional [公式] to the subspace [公式] such that [公式] for all [公式] .

Proof: [公式] must of course also be nonzero. Let [公式] be our extended functional, with [公式] and [公式] . We require that for any [公式] , for any [公式] ,

[公式] The case of [公式] is trivial.

In the case of [公式] , the inequality in [公式] is equivalent to

[公式] In the case of [公式] , the inequality in [公式] is equivalent to

[公式]

Here, we notice that the product of any [公式] by any scalar is also in [公式] , by the closure property of subspace. Thus, if we show that for arbitrary [公式] ,

[公式] we have shown the existence of the desired [公式] . [公式] follows from

[公式]

in which we used linearity of [公式] , the fact that [公式] on [公式] , and the triangle inequality on [公式] . This completes our proof. [公式]

The proof of the Hahn-Banach theorem, of which Lemma 1, is the most difficulty part, can come across as coming out of the blue. I certainly developed a better idea of how to derive it by “working backwards”, as done above, first assuming the existence of the desired property, then finding a condition that implies it, and finally proving that that condition is indeed satisfied.

Theorem 1 (Hahn Banach theorem) Let [公式] be a real vector space on which is defined a sublinear functional [公式] . Let [公式] be any subspace of it and [公式] be some linear functional such that [公式] for all [公式] . Then, there exists a linear functional [公式] such that for all [公式] , [公式] and for all [公式] , [公式] .

Proof: Lemma 1 tells us that if [公式] , we can always extend [公式] onto some subspace [公式], which is a proper extension of subspace [公式] such the extension of [公式] is bounded above by [公式] on [公式] and agrees with [公式] on [公式] . Let [公式] be the collection of two-tuples [公式] such that [公式] is a linear functional defined on [公式] and bounded above by [公式] on [公式] . Let [公式] . We say that [公式] iff [公式] and [公式] on [公式] . One easy verifies that this is a partial order on [公式] . For any chain in [公式] , we take the union of all sets in the chain [公式] and define a function [公式] with for any [公式] , [公式] for some [公式] in the chain that is defined on a domain that contains [公式] . It is apparent that for any [公式] in the chain, [公式] . Thus, we can apply Zorn’s lemma to derive the existence of a maximal element in [公式] with respect to this partial order. The set associated with any maximal element must be [公式] itself in order for Lemma 1 to not be violated. [公式]

Now, we will go about generalized the Hahn-Banach theorem to complex vector spaces.

Lemma 2 Let [公式] be a complex vector space and let [公式] be a linear functional on [公式] . If [公式] is a complex linear functional on [公式] and [公式] , then [公式] is a real linear functional, and [公式] for all [公式] . Conversely, if [公式] is a real linear functional on [公式] and [公式] is defined by [公式] , then [公式] is complex linear. In this case, if [公式] is normed, we have [公式] .

Proof: Let [公式] Then, for any [公式], we write [公式] , where [公式]. We have [公式] and [公式] . Thus, [公式] . For any [公式] , [公式] . That for any [公式] , [公式] is also easily verified.

For the converse, one easily verifies that [公式], and for [公式] ,

[公式] For any [公式] , we have that [公式] . This shows that [公式] . With [公式] for some [公式] , [公式] , since [公式] is linear, we have that [公式] . This shows that for any [公式] , there exists a [公式] of the same norm such that [公式] , which shows that [公式] . This completes our proof. [公式]

In the proof of the above lemma, we omitted the case of [公式] . [1] introduced the notation

[公式]

Using this we can define the polar decomposition of any [公式] as

[公式] Applying to [公式] gives us [公式] . We note that in the proof of Lemma 2, we multiplied by [公式] .

Lemma 3 For any complex vector space [公式] , there exists a real vector space [公式] and a function [公式] that is bijective and linear with respect to real but not complex coefficients.

Proof: For any [公式] , we must have [公式] and also [公式] for all [公式] . We also stipulate that for any [公式] , [公式] .Take any basis [公式] of [公式] . Then we have as a set of basis elements defining [公式] , [公式] . To verify that no non-trivial linear combination of basis elements of [公式] can equal [公式] , one can simply use linearity to derive violation of the definition of basis [公式] of [公式] in the case of linear dependence of a subset of basis elements of [公式] .

We have defined [公式] to be linear with respect to real coefficients. It is not at all linear with respect to complex coefficients as [公式] and [公式] are basis elements of a real vector space; since the underlying field is [公式] multiplying a vector of it by an imaginary number is simply not defined here. [公式]

Theorem 2 (Complex Hahn-Banach Theorem) Let [公式] be a complex vector space, [公式] a seminorm on [公式] , [公式] a subspace of [公式] , and [公式] a complex linear functional on [公式] such that [公式] for [公式] . Then there exists a complex linear functional [公式] on [公式] such that [公式] for all [公式] and [公式]

Proof: Let [公式] . By Lemma 3, there exists a real vector space [公式] with [公式] linear with respect to real coefficients and bijective. Let [公式] be defined by [公式] . One easily verifies that [公式] is real linear functional on [公式] . Moreover, on [公式] , which is a subspace of [公式], [公式] , with [公式] a seminorm on [公式] easily verified as well. By the Hahn-Banach theorem for real vector spaces (Theorem 1), there is a real linear functional [公式] defined on [公式] that agrees with [公式] on [公式] such that for all [公式] , [公式] . From this we also derive an analogous extension of [公式] to [公式] , which is [公式] . Now, let [公式] on [公式] . As in the proof of Lemma 2, if [公式] , we have [公式] . Since [公式] on [公式] , we also have [公式] on [公式] . This completes our proof. [公式]

Theorem 3 Let [公式] be a normed vector space (over [公式] ). Then,

  1. If [公式] is a subspace of [公式] and [公式] , there exists [公式] such that [公式] and [公式] . In fact, if [公式] , [公式] can be defined such that [公式] , in which case [公式] .
  2. If [公式] , there exists [公式] such that [公式] and [公式] .
  3. The bounded linear functionals on [公式] separate points.

Proof: We wish to define [公式] on [公式] , which is, by linearity, done simply by prescribing the value of [公式] . The function [公式] on [公式] is such that [公式] if and only if [公式] . We need to prescribe an [公式] and we try the largest value which satisfies the requirement as given in the Hahn-Banach theorem, where the take [公式] to be the upper bounding seminorm. We let [公式] . Take arbitrary [公式] . We now wish to show that [公式]

The inequality in this is equivalent to

[公式] which is true from the definition of [公式] .

We have by the definition of operator norm that

[公式] Set [公式] and pick a sequence [公式] such that [公式] from above which exists by definition of [公式] . We have thus shown that [公式] . Obviously, from this we also have [公式] . Applying Hahn-Banach theorem to [公式] with [公式] as the seminorm to the extension of [公式] shows that we can extend the extension of all of [公式] , which completes the proof of (1).

(2) is a special case of (1) with [公式] . As for (3), if [公式] , ether [公式] is a complex multiple of [公式] or not. If yes, [公式] would suffice. If not, we can define [公式] on [公式] to be the constant zero function, and then, by (1), we can extend [公式] to a function [公式] on the entire space such that [公式] . [公式]

References

  • [1] Gerald B. Folland. Real Analysis – Modern Techniques and their Applications. John Wiley & Sons, Inc., 1999.

On normed vector spaces

Originally published at 狗和留美者不得入内. You can comment here or there.

Here, the underlying field [公式] of any vector space shall be either [公式] or [公式] . Moreover, subspace will always denote the subspace of a vector space.

Definition 1 A seminorm on a vector space [公式] over [公式] is a function [公式] that satisfies the following properties.

  1. Absolute homogeneity: for all [公式] , [公式] , [公式] .
  2. Triangle inequality: for all [公式] , [公式] .

Proposition 1 For any seminorm [公式] , [公式] .

Proof: Follows directly from absolute homogeneity. [公式]

Definition 2 A norm [公式] on a vector space is a seminorm such that [公式] iff [公式] .

Definition 3 A vector space equipped with a norm is called a normed vector space. The topology it defines is called the norm topology on [公式]

Definition 4 A sequence of vectors [公式] in vector space [公式] converges with respect to norm [公式] iff [公式] .

Definition 5 A normed vector space that is complete with respect to the norm metric is called a Banach space.

Definition 6 A series [公式] converges absolutely iff [公式] .

Theorem 1 A normed vector space [公式] is complete iff every series in it that converges absolutely also converges with respect to the norm topology.

Proof: We assume the space is complete. This means that for any Cauchy sequence [公式] , [公式] for some [公式] . Now take any [公式] such that [公式] , which of course means that [公式] . To show that it converges, it suffices to show that [公式] is Cauchy. We have that for all

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<p><small>Originally published at <a href="https://gmachine1729.wpcomstaging.com/2021/06/10/on-normed-vector-spaces/">狗和留美者不得入内</a>. You can comment here or <a href="https://gmachine1729.wpcomstaging.com/2021/06/10/on-normed-vector-spaces/#comments">there</a>.</small></p><div class="RichText ztext Post-RichText"> <p>Here, the underlying field <img src="https://www.zhihu.com/equation?tex=K" alt="[公式]" eeimg="1" data-formula="K"> of any vector space shall be either <img src="https://www.zhihu.com/equation?tex=%5Cmathbb%7BR%7D" alt="[公式]" eeimg="1" data-formula="\mathbb{R}"> or <img src="https://www.zhihu.com/equation?tex=%5Cmathbb%7BC%7D" alt="[公式]" eeimg="1" data-formula="\mathbb{C}"> . Moreover, subspace will always denote the subspace of a vector space.</p> <p><b>Definition 1</b> A <i>seminorm</i> on a vector space <img src="https://www.zhihu.com/equation?tex=X" alt="[公式]" eeimg="1" data-formula="X"> over <img src="https://www.zhihu.com/equation?tex=K" alt="[公式]" eeimg="1" data-formula="K"> is a function <img src="https://www.zhihu.com/equation?tex=%5Cparallel+%5Ccdot+%5Cparallel+%3A+X+%5Cto+%5B0%2C+%5Cinfty%29" alt="[公式]" eeimg="1" data-formula="\parallel \cdot \parallel : X \to [0, \infty)"> that satisfies the following properties.</p> <ol> <li>Absolute homogeneity: for all <img src="https://www.zhihu.com/equation?tex=x+%5Cin+X" alt="[公式]" eeimg="1" data-formula="x \in X"> , <img src="https://www.zhihu.com/equation?tex=%5Clambda+%5Cin+K" alt="[公式]" eeimg="1" data-formula="\lambda \in K"> , <img src="https://www.zhihu.com/equation?tex=%5Cparallel+%5Clambda+x%5Cparallel+%3D+%7C%5Clambda%7C+%5Cparallel+x%5Cparallel" alt="[公式]" eeimg="1" data-formula="\parallel \lambda x\parallel = |\lambda| \parallel x\parallel"> .</li> <li>Triangle inequality: for all <img src="https://www.zhihu.com/equation?tex=x%2Cy%5Cin+X" alt="[公式]" eeimg="1" data-formula="x,y\in X"> , <img src="https://www.zhihu.com/equation?tex=%5Cparallel+x%2By%5Cparallel+%5Cleq+%5Cparallel+x+%5Cparallel+%2B+%5Cparallel+y+%5Cparallel" alt="[公式]" eeimg="1" data-formula="\parallel x+y\parallel \leq \parallel x \parallel + \parallel y \parallel"> .</li> </ol> <p><b>Proposition 1</b> For any seminorm <img src="https://www.zhihu.com/equation?tex=%5Cparallel+%5Ccdot+%5Cparallel" alt="[公式]" eeimg="1" data-formula="\parallel \cdot \parallel"> , <img src="https://www.zhihu.com/equation?tex=%5Cparallel+0%5Cparallel+%3D+0" alt="[公式]" eeimg="1" data-formula="\parallel 0\parallel = 0"> .</p> <p><i>Proof</i>: Follows directly from absolute homogeneity. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" eeimg="1" data-formula="\square"> </p> <p><b>Definition 2</b> A <i>norm</i> <img src="https://www.zhihu.com/equation?tex=%5Cparallel+%5Ccdot+%5Cparallel" alt="[公式]" eeimg="1" data-formula="\parallel \cdot \parallel"> on a vector space is a <i>seminorm</i> such that <img src="https://www.zhihu.com/equation?tex=%5Cparallel+x+%5Cparallel+%3D+0" alt="[公式]" eeimg="1" data-formula="\parallel x \parallel = 0"> iff <img src="https://www.zhihu.com/equation?tex=x+%3D+0" alt="[公式]" eeimg="1" data-formula="x = 0"> .</p> <p><b>Definition 3</b> A vector space equipped with a norm is called a <i>normed vector space</i>. The topology it defines is called the <i>norm topology</i> on <img src="https://www.zhihu.com/equation?tex=X" alt="[公式]" eeimg="1" data-formula="X"> </p> <p><b>Definition 4</b> A sequence of vectors <img src="https://www.zhihu.com/equation?tex=%5C%7Bx_n%5C%7D" alt="[公式]" eeimg="1" data-formula="\{x_n\}"> in vector space <img src="https://www.zhihu.com/equation?tex=X" alt="[公式]" eeimg="1" data-formula="X"> converges with respect to norm <img src="https://www.zhihu.com/equation?tex=%5Cparallel+%5Ccdot+%5Cparallel" alt="[公式]" eeimg="1" data-formula="\parallel \cdot \parallel"> iff <img src="https://www.zhihu.com/equation?tex=%5Cparallel+x_n+-+x%5Cparallel+%5Cto+0" alt="[公式]" eeimg="1" data-formula="\parallel x_n - x\parallel \to 0"> .</p> <p><b>Definition 5</b> A normed vector space that is complete with respect to the norm metric is called a <i>Banach space</i>.</p> <p><b>Definition 6</b> A series <img src="https://www.zhihu.com/equation?tex=%5Csum_%7Bi%3D1%7D%5E%5Cinfty+x_i" alt="[公式]" eeimg="1" data-formula="\sum_{i=1}^\infty x_i"> <i>converges absolutely</i> iff <img src="https://www.zhihu.com/equation?tex=%5Csum_%7Bi%3D1%7D%5E%5Cinfty+%5Cparallel+x_i%5Cparallel+%3C+%5Cinfty" alt="[公式]" eeimg="1" data-formula="\sum_{i=1}^\infty \parallel x_i\parallel < \infty"> .</p> <p><b>Theorem 1</b> A normed vector space <img src="https://www.zhihu.com/equation?tex=X" alt="[公式]" eeimg="1" data-formula="X"> is complete iff every series in it that converges absolutely also converges with respect to the norm topology.</p> <p><i>Proof</i>: We assume the space is complete. This means that for any Cauchy sequence <img src="https://www.zhihu.com/equation?tex=%5C%7Bx_n%5C%7D" alt="[公式]" eeimg="1" data-formula="\{x_n\}"> , <img src="https://www.zhihu.com/equation?tex=x_n+%5Cto+x" alt="[公式]" eeimg="1" data-formula="x_n \to x"> for some <img src="https://www.zhihu.com/equation?tex=x+%5Cin+X" alt="[公式]" eeimg="1" data-formula="x \in X"> . Now take any <img src="https://www.zhihu.com/equation?tex=%5C%7By_n%5C%7D" alt="[公式]" eeimg="1" data-formula="\{y_n\}"> such that <img src="https://www.zhihu.com/equation?tex=%5Csum_%7Bi%3D1%7D%5E%5Cinfty+%5Cparallel+y_i%5Cparallel+%3C+%5Cinfty" alt="[公式]" eeimg="1" data-formula="\sum_{i=1}^\infty \parallel y_i\parallel < \infty"> , which of course means that <img src="https://www.zhihu.com/equation?tex=%5Cparallel+y_n+%5Cparallel+%5Cto+0" alt="[公式]" eeimg="1" data-formula="\parallel y_n \parallel \to 0"> . To show that it converges, it suffices to show that <img src="https://www.zhihu.com/equation?tex=S_n+%3D+%5Csum_%7Bi%3D1%7D%5En+y_i" alt="[公式]" eeimg="1" data-formula="S_n = \sum_{i=1}^n y_i"> is Cauchy. We have that for all <img src="https://www.zhihu.com/equation?tex=%5Cepsilon%3E0" alt="[公式]" eeimg="1" data-formula="\epsilon>0&#8243;> , there exists <img src="https://www.zhihu.com/equation?tex=N" alt="[公式]" eeimg="1" data-formula="N"> such that <img src="https://www.zhihu.com/equation?tex=n+%5Cgeq+m%5Cgeq+N" alt="[公式]" eeimg="1" data-formula="n \geq m\geq N"> implies <img src="https://www.zhihu.com/equation?tex=%5Csum_%7Bi%3Dm%7D%5En+%5Cparallel+y_i%5Cparallel+%3C+%5Cepsilon" alt="[公式]" eeimg="1" data-formula="\sum_{i=m}^n \parallel y_i\parallel < \epsilon"> . The triangle inequality implies that <img src="https://www.zhihu.com/equation?tex=%5Cleft%5ClVert+%5Csum_%7Bi%3Dm%7D%5En+y_i+%5Cright%5CrVert+%3C+%5Cepsilon" alt="[公式]" eeimg="1" data-formula="\left\lVert \sum_{i=m}^n y_i \right\rVert < \epsilon"> , which shows that <img src="https://www.zhihu.com/equation?tex=S_n" alt="[公式]" eeimg="1" data-formula="S_n"> is Cauchy.</p> <p>We now assume that absolute convergence implies convergence with respect to norm topology. Take any Cauchy sequence <img src="https://www.zhihu.com/equation?tex=%5C%7By_n%5C%7D" alt="[公式]" eeimg="1" data-formula="\{y_n\}"> . For all <img src="https://www.zhihu.com/equation?tex=k+%5Cin+%5Cmathbb%7BN%7D" alt="[公式]" eeimg="1" data-formula="k \in \mathbb{N}"> , there exists a minimum <img src="https://www.zhihu.com/equation?tex=n_k" alt="[公式]" eeimg="1" data-formula="n_k"> such that <img src="https://www.zhihu.com/equation?tex=n+%5Cgeq+m+%5Cgeq+n_k" alt="[公式]" eeimg="1" data-formula="n \geq m \geq n_k"> implies <img src="https://www.zhihu.com/equation?tex=%5ClVert+y_n+-+y_%7Bm%7D+%5CrVert+%3C+%5Cfrac%7B1%7D%7B2%5Ek%7D" alt="[公式]" eeimg="1" data-formula="\lVert y_n - y_{m} \rVert < \frac{1}{2^k}"> . From this we derive a subsequence <img src="https://www.zhihu.com/equation?tex=%5C%7By_%7Bn_k%7D%5C%7D" alt="[公式]" eeimg="1" data-formula="\{y_{n_k}\}"> . Let <img src="https://www.zhihu.com/equation?tex=%5C%7Bx_k+%3D+y_%7Bn_%7Bk%2B1%7D%7D+-+y_%7Bn_%7Bk%7D%7D%5C%7D" alt="[公式]" eeimg="1" data-formula="\{x_k = y_{n_{k+1}} - y_{n_{k}}\}"> . We have that <img src="https://www.zhihu.com/equation?tex=y_%7Bn_%7Bk%2B1%7D%7D+%3D+y_%7Bn_1%7D%2B%5Csum_%7Bi%3D1%7D%5Ek+x_i" alt="[公式]" eeimg="1" data-formula="y_{n_{k+1}} = y_{n_1}+\sum_{i=1}^k x_i"> . That upper bound by <img src="https://www.zhihu.com/equation?tex=%5Cfrac%7B1%7D%7B2%5Ek%7D" alt="[公式]" eeimg="1" data-formula="\frac{1}{2^k}"> tells us that <img src="https://www.zhihu.com/equation?tex=%5Csum_%7Bi%3D1%7D%5E%5Cinfty+%5ClVert+x_k+%5CrVert+%3C+%5Cinfty" alt="[公式]" eeimg="1" data-formula="\sum_{i=1}^\infty \lVert x_k \rVert < \infty"> , which by our hypothesis implies that <img src="https://www.zhihu.com/equation?tex=%5Csum_%7Bi%3D1%7D%5E%5Cinfty+x_k" alt="[公式]" eeimg="1" data-formula="\sum_{i=1}^\infty x_k"> is convergent, which tells us that <img src="https://www.zhihu.com/equation?tex=%5C%7By_%7Bn_k%7D%5C%7D" alt="[公式]" eeimg="1" data-formula="\{y_{n_k}\}"> is convergent to the same value. Its being a subsequence of Cauchy sequence <img src="https://www.zhihu.com/equation?tex=%5C%7By_n%5C%7D" alt="[公式]" eeimg="1" data-formula="\{y_n\}"> means that <img src="https://www.zhihu.com/equation?tex=%5C%7By_n%5C%7D" alt="[公式]" eeimg="1" data-formula="\{y_n\}"> converges to the same value too. This completes our proof. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" eeimg="1" data-formula="\square"> </p> <p>This proposition felt initially elusive or not very intuitive to me. However, once one realize that one can telescope a Cauchy sequence to express it as a sequence of partial sums, it is natural to realize that the condition regarding absolute convergence would imply completeness.</p> <p><b>Definition 7</b> Let <img src="https://www.zhihu.com/equation?tex=X%2C+Y" alt="[公式]" eeimg="1" data-formula="X, Y"> be two normed vector spaces. Let <img src="https://www.zhihu.com/equation?tex=A%3A+X+%5Cto+Y" alt="[公式]" eeimg="1" data-formula="A: X \to Y"> be a linear operator. Moreover, the following are equivalent.</p> <ol> <li><img src="https://www.zhihu.com/equation?tex=A" alt="[公式]" eeimg="1" data-formula="A"> is <i>bounded</i>.</li> <li>There exists <img src="https://www.zhihu.com/equation?tex=C+%5Cin+%5Cmathbb%7BR%7D" alt="[公式]" eeimg="1" data-formula="C \in \mathbb{R}"> such that for all <img src="https://www.zhihu.com/equation?tex=x+%5Cin+X" alt="[公式]" eeimg="1" data-formula="x \in X"> ,<img src="https://www.zhihu.com/equation?tex=%5ClVert+Ax%5CrVert+%5Cleq+C%5ClVert+x%5CrVert" alt="[公式]" eeimg="1" data-formula="\lVert Ax\rVert \leq C\lVert x\rVert"> .</li> <li>For some <img src="https://www.zhihu.com/equation?tex=%5Cdelta+%3E+0" alt="[公式]" eeimg="1" data-formula="\delta > 0&#8243;> , there exists <img src="https://www.zhihu.com/equation?tex=C+%5Cin+%5Cmathbb%7BR%7D" alt="[公式]" eeimg="1" data-formula="C \in \mathbb{R}"> such that for all <img src="https://www.zhihu.com/equation?tex=x+%5Cin+X" alt="[公式]" eeimg="1" data-formula="x \in X"> such that <img src="https://www.zhihu.com/equation?tex=%5ClVert+x%5CrVert+%3D+%5Cdelta" alt="[公式]" eeimg="1" data-formula="\lVert x\rVert = \delta"> ,<img src="https://www.zhihu.com/equation?tex=%5ClVert+Ax%5CrVert+%5Cleq+C%5ClVert+x%5CrVert" alt="[公式]" eeimg="1" data-formula="\lVert Ax\rVert \leq C\lVert x\rVert"> .</li> </ol> <p><b>Proposition 2</b> Let <img src="https://www.zhihu.com/equation?tex=X%2C+Y" alt="[公式]" eeimg="1" data-formula="X, Y"> be two normed vector spaces. Let <img src="https://www.zhihu.com/equation?tex=A%3A+X+%5Cto+Y" alt="[公式]" eeimg="1" data-formula="A: X \to Y"> be a linear operator. The following are equivalent.</p> <ol> <li><img src="https://www.zhihu.com/equation?tex=A" alt="[公式]" eeimg="1" data-formula="A"> is continuous.</li> <li><img src="https://www.zhihu.com/equation?tex=A" alt="[公式]" eeimg="1" data-formula="A"> is continuous at <img src="https://www.zhihu.com/equation?tex=0" alt="[公式]" eeimg="1" data-formula="0">. </li> <li><img src="https://www.zhihu.com/equation?tex=A" alt="[公式]" eeimg="1" data-formula="A"> is bounded.</li> </ol> <p><i>Proof</i>: That (1) implies (2) is immediate. Assume (2), which means that every neighborhood <img src="https://www.zhihu.com/equation?tex=N_Y" alt="[公式]" eeimg="1" data-formula="N_Y"> of <img src="https://www.zhihu.com/equation?tex=0+%5Cin+Y" alt="[公式]" eeimg="1" data-formula="0 \in Y"> , there exists an open ball of radius <img src="https://www.zhihu.com/equation?tex=%5Cdelta" alt="[公式]" eeimg="1" data-formula="\delta"> centered at <img src="https://www.zhihu.com/equation?tex=0+%5Cin+X" alt="[公式]" eeimg="1" data-formula="0 \in X"> , which we denote via <img src="https://www.zhihu.com/equation?tex=B%280%2C+%5Cdelta%29" alt="[公式]" eeimg="1" data-formula="B(0, \delta)"> , such that <img src="https://www.zhihu.com/equation?tex=A%28B%280%2C+%5Cdelta%29%29+%5Csubset+N_Y" alt="[公式]" eeimg="1" data-formula="A(B(0, \delta)) \subset N_Y"> . Let <img src="https://www.zhihu.com/equation?tex=N_Y" alt="[公式]" eeimg="1" data-formula="N_Y"> be bounded above in norm by <img src="https://www.zhihu.com/equation?tex=M+%3E+0" alt="[公式]" eeimg="1" data-formula="M > 0&#8243;> . Then, (3) of Definition 7 is satisfied, or equivalently, <img src="https://www.zhihu.com/equation?tex=A" alt="[公式]" eeimg="1" data-formula="A"> is bounded. Now we show that (3) implies (1). Assume (3), namely that there exists <img src="https://www.zhihu.com/equation?tex=C%3E0" alt="[公式]" eeimg="1" data-formula="C>0&#8243;> such that <img src="https://www.zhihu.com/equation?tex=%5ClVert+Ax%5CrVert+%5Cleq+C+%5ClVert+x%5CrVert" alt="[公式]" eeimg="1" data-formula="\lVert Ax\rVert \leq C \lVert x\rVert"> for all <img src="https://www.zhihu.com/equation?tex=x+%5Cin+X" alt="[公式]" eeimg="1" data-formula="x \in X"> . In that case, if <img src="https://www.zhihu.com/equation?tex=%5ClVert+x_1+-+x_2+%5CrVert+%3C+C%5E%7B-1%7D%5Cepsilon" alt="[公式]" eeimg="1" data-formula="\lVert x_1 - x_2 \rVert < C^{-1}\epsilon"> , then <img src="https://www.zhihu.com/equation?tex=%5ClVert+Ax_1+-+Ax_2%5CrVert+%3D+%5ClVert+A%28x_1+-+x_2%29+%5CrVert+%3C+%5Cepsilon" alt="[公式]" eeimg="1" data-formula="\lVert Ax_1 - Ax_2\rVert = \lVert A(x_1 - x_2) \rVert < \epsilon"> . This implies that if <img src="https://www.zhihu.com/equation?tex=%5C%7Bx_n%5C%7D" alt="[公式]" eeimg="1" data-formula="\{x_n\}"> is Cauchy than <img src="https://www.zhihu.com/equation?tex=%5C%7BAx_n%5C%7D" alt="[公式]" eeimg="1" data-formula="\{Ax_n\}"> is also Cauchy, which is a definition of continuity in a metric space. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" eeimg="1" data-formula="\square"> </p> <p><b>Definition 8</b> We define a function <img src="https://www.zhihu.com/equation?tex=T+%5Cmapsto+%5ClVert+T%5CrVert" alt="[公式]" eeimg="1" data-formula="T \mapsto \lVert T\rVert"> by</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%5ClVert+T%5CrVert+%26%3D%26+%5Csup%5C%7B%5ClVert+Tx%5CrVert%3A+%5ClVert+x%5CrVert+%3D+1%5C%7D%5C%5C+%26%3D%26%5Csup%5Cleft%5C%7B%5Cfrac%7B%5ClVert+Tx%5CrVert%7D%7B%5ClVert+x%5CrVert%7D%3A+x+%5Cneq+0%5Cright%5C%7D%5C%5C+%26%3D%26+%5Cinf+%5C%7BC+%3A+%5ClVert+Tx%5CrVert+%5Cleq+C%5ClVert+x%5CrVert%2C+%5Cforall+x%5C%7D+%5Cend%7Beqnarray%7D%5C%5C" alt="[公式]" eeimg="1" data-formula="\begin{eqnarray} \lVert T\rVert &amp;=&amp; \sup\{\lVert Tx\rVert: \lVert x\rVert = 1\}\\ &amp;=&amp;\sup\left\{\frac{\lVert Tx\rVert}{\lVert x\rVert}: x \neq 0\right\}\\ &amp;=&amp; \inf \{C : \lVert Tx\rVert \leq C\lVert x\rVert, \forall x\} \end{eqnarray}\\"> </p> <p>on the vector space <img src="https://www.zhihu.com/equation?tex=%5Cmathcal%7BL%7D%28X%2CY%29" alt="[公式]" eeimg="1" data-formula="\mathcal{L}(X,Y)"> of linear transformations from <img src="https://www.zhihu.com/equation?tex=X" alt="[公式]" eeimg="1" data-formula="X"> to <img src="https://www.zhihu.com/equation?tex=Y" alt="[公式]" eeimg="1" data-formula="Y"> , which is called the <i>operator norm</i>. We leave to the reader to verify that it is a norm.</p> <p><b>Proposition 3</b> If <img src="https://www.zhihu.com/equation?tex=Y" alt="[公式]" eeimg="1" data-formula="Y"> is complete, so is <img src="https://www.zhihu.com/equation?tex=%5Cmathcal%7BL%7D%28X%2CY%29" alt="[公式]" eeimg="1" data-formula="\mathcal{L}(X,Y)"> .</p> <p><i>Proof</i>: For any Cauchy sequence <img src="https://www.zhihu.com/equation?tex=%5C%7BA_n%5C%7D" alt="[公式]" eeimg="1" data-formula="\{A_n\}"> in <img src="https://www.zhihu.com/equation?tex=%5Cmathcal%7BL%7D%28X%2CY%29" alt="[公式]" eeimg="1" data-formula="\mathcal{L}(X,Y)"> , <img src="https://www.zhihu.com/equation?tex=%5C%7BA_n+x%5C%7D" alt="[公式]" eeimg="1" data-formula="\{A_n x\}"> is also Cauchy. Thus, we can define <img src="https://www.zhihu.com/equation?tex=A%3A+X+%5Cto+Y" alt="[公式]" eeimg="1" data-formula="A: X \to Y"> by <img src="https://www.zhihu.com/equation?tex=Ax+%3D+%5Clim+A_n+x" alt="[公式]" eeimg="1" data-formula="Ax = \lim A_n x"> . We leave to the reader to verify that <img src="https://www.zhihu.com/equation?tex=A+%5Cin+%5Cmathcal%7BL%7D%28X%2CY%29" alt="[公式]" eeimg="1" data-formula="A \in \mathcal{L}(X,Y)"> and that it is indeed in limit with respect to the operator norm. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" eeimg="1" data-formula="\square"> </p> <p><b>Proposition 4</b> If <img src="https://www.zhihu.com/equation?tex=B+%5Cin+%5Cmathcal%7BL%7D%28X%2CY%29%2C+A+%5Cin+%5Cmathcal%7BL%7D%28Y%2CZ%29" alt="[公式]" eeimg="1" data-formula="B \in \mathcal{L}(X,Y), A \in \mathcal{L}(Y,Z)"> , then</p> <p><img src="https://www.zhihu.com/equation?tex=%5ClVert+AB%5CrVert+%5Cleq+%5ClVert+A%5ClVert+%5ClVert+B%5ClVert.%5C%5C+" alt="[公式]" eeimg="1" data-formula="\lVert AB\rVert \leq \lVert A\lVert \lVert B\lVert.\\ "> </p> <p><i>Proof</i>: Very mechanical and left to the reader. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" eeimg="1" data-formula="\square"> </p> <p><b>References</b></p> <ul> <li>[1] Gerald B. Folland. <i>Real Analysis &#8211; Modern Techniques and their Applications</i>. John Wiley &amp; Sons, Inc., 1999.</li> </ul> </div>

关于曲线的长度和曲率

Originally published at 狗和留美者不得入内. You can comment here or there.

找到了一本俄罗斯人写的微分几何教材,当初看了觉得太枯燥,但近几天当弥补了不少自己的微积分和分析基础后却开始学进去了。我写的英语的数学文章的结果大多都是细节自己独立重证了。如果基本直接抄书,那写的意义可能也没那么大。我想如果我要抄书,那就把它翻译成中文吧,顺便也把中文的名词也给学会。对关于曲线的曲率的那段,我现在就准备这样做,当然如我当时翻译朗道的教科书一样,有的时候也会加一点我自己的想法或补点细节。

讲曲率之前,得先讲讲曲线的长度。

曲线的长度

[公式] 为某个曲线的闭弧, [公式] 为其参数化; [公式] 。我们注意到一个多边形线[公式][公式] )的一个由穿过某有序的有限的点集合 [公式] 的相邻的点的线段构成的曲线。一个多边形线 [公式] 是一个正则内接于曲线 [公式] 的多边形当存在线段 [公式] 的以点 [公式] 的满足 [公式] 的分割 [公式] 。对每个多边形线对应其长度 [公式] 。我们以 [公式] 标记所有正则内接于曲线 [公式] 的多边形线的集合

定义 1.4.1 一个连续曲线 [公式][公式] 被称为可求长曲线

定义 1.4.2 可求长曲线 [公式]长度定义为 [公式]

定理 1.4.1 光滑曲线的闭弧是可求长的,其长度为

[公式]

证明:相当繁琐的用到区间分割的不等式估计。为了时间的考虑暂时不过。此证明也大概率类似于黎曼积分的重要定理的证明。

任意曲线若其所有闭弧都是可求长被称为可求长曲线。对可求长曲线,可以定义基于每一个闭弧的长度的存在的所谓的弧长参数化。取任意点 [公式] 并联于 [公式] 参数 [公式] 的零值。为任意其他点 [公式] ,对应于等于的 [公式] 的弧长 [公式] 参数的值,若 [公式][公式] 之后我们给予其正符号 [公式] ,若 [公式][公式] 之前,我们给予其负符号 [公式] 。若 [公式] 有个光滑正则参数化 [公式] ,其弧长参数化也是光滑的,正则的。当考虑到符号,我们推导出弧长 [公式] 。函数 [公式] 是可导的, [公式] 。从而,存在反函数 [公式] ,其导数为

[公式]

曲线[公式] 的弧长度(或单元速度)参数化定义于公式

[公式][公式] ,我们得到矢量函数 [公式] 的可导,其导数为

[公式] 最后一个公式表明此弧长参数化的正则的。以弧长参数化 [公式] 来表示,切矢量 [公式] ,主法矢量 [公式] 和副法矢量 [公式] 的形式简单如下:

[公式]

第一个公式由 [公式] 成立,第二个成立于等式

[公式] 从这个,我们得到了 [公式] 之间的正交。最后一个公式成立于矢量 [公式] 的定义。

曲线的曲率

[公式][公式] 里的一个光滑的曲线。在此取一个点 [公式] ,另一个点 [公式] 。我们以 [公式] 标记在 [公式][公式] 的弧长,以 [公式] 标记 [公式][公式][公式] 切矢量 [公式][公式]

定义 1.6.1 极限

[公式]

若存在,叫做曲线 [公式] 在点 [公式]曲率

我们会将曲线 [公式] 在点 [公式] 的曲率标记以 [公式]

例子 1.6.1 (a) 若 [公式] 是直线,在 [公式] 的所有点,[公式][公式] 。(b) 若 [公式] 是半径为 [公式] 的圆,很容易得到圆的所有点的曲率都是 [公式]

定理 1.6.1[公式][公式] 正则曲线。在其所有点都有曲率。若 [公式] 是个 [公式] 的正则参数化,则 [公式] .
证明[公式][公式] 的弧长参数化,令 [公式] 。从而, [公式][公式] 为矢量 [公式] 之间的角度。由于 [公式][公式] ,故

[公式]

以这些,我们证明了定理的断言曲率存在的第一部分,并得以公式

[公式][公式] 为任意 [公式] 的正则参数化。利用 [公式] 做个计算会得到

[公式]

定理 1.6.2 在一个 [公式] 正则曲线 [公式] 的任意的点,密切平面的唯一存在的必要并且足够的条件是 [公式] 的曲率在此点不等于零。

证明:从定理1.6.1,我们可以看到曲率不等于零当且仅当 [公式] 之间不是平行的。在这种情况下,根据 [2] 里的 Theorem 2 的证明,只有一个密切平面存在。当曲率等于零时, [公式] ,故 [公式][公式] ,一个直线。显然,任何包括直线的(多个)平面都是其密切平面。 [公式]

定理 1.6.3 我们假设某个嵌入三维空间的平面包含所有曲线的点。令 [公式] 为任意在线段 [公式] 上的连续函数。曲率函数为 [公式][公式] 为弧长参数的曲线 [公式] 有唯一的存在(在由刚性运动而定义的等价关系下)。

证明:我们想寻求满足

[公式] 的函数 [公式] 。当我们不失去一般性地假设平面的法矢量为 [公式] ,从公式 [公式] ,我们能得到

[公式] 然后不难发觉当

[公式] 我们会得到

[公式][公式] 会给我们

[公式]

[公式] 为在初始点的切矢量 [公式][公式] 轴的角度。两个曲率一致的曲线的切矢量的变化也是一致的,则之间的切矢量的差是常矢量,位置的差也是长矢量。我们主要到在这里,切矢量的范数皆为 [公式] 。很明显当此俩曲率同等,可以将坐标轴做刚性变换(旋转,反射或平移)而将某一个曲线映射到另一个。证明的剩下细节留给阅读者。 [公式]

习题 1.7.3 (平面曲线的Frenet公式)证明公式

[公式] 同等于等式

[公式]

:一个简单的计算。

References

On the tangent line and osculating plane of a curve

Originally published at 狗和留美者不得入内. You can comment here or there.

Here, we will be working in [公式] .

Analytic geometry prerequisites

Proposition 1 The distance between a point [公式] and the plane given by [公式] is [公式] .

Proof: A normal vector of the plane is [公式] . We plug in [公式] to get

[公式] the solution of which is [公式] . Since every unit of [公式] corresponds to [公式] of distance, we have for our answer[公式]. [公式]

Proposition 2 The distance between a point [公式] and a straight line given by [公式] can be obtained by the magnitude of a cross product.

Proof: As for this distance, it is obtained by taking the perpendicular with respect the straight line that contains [公式] , which we shall call [公式] . We use [公式] to denote the distance between [公式] and [公式] . One notices that [公式] is equal to [公式] , where [公式] is the angle between the straight line given in the proposition and the straight line connecting [公式] and [公式] . We know that the magnitude of the cross product of two vectors is the product of their magnitudes and the sign of the angle between the two vectors, which completes our proof. [公式]

Preliminary definitions

Definition 1 A regular curve is a connected subset [公式] of [公式] homeomorphic to some [公式] that is a line segment [公式] or a circle of radius [公式] . If the homeomorphism [公式] is in [公式] for [公式] and the rank of [公式] is maximal (equal to 1), then we say this curve is k-fold continuously differentiable. For [公式] , we say that [公式] is smooth.

Definition 2 Let a smooth curve [公式] be given by the parametric equations

[公式] The velocity vector of [公式] at [公式] is the derivative

[公式]The velocity vector field is the vector function [公式] . The speed of [公式] at [公式] is the length [公式] of the velocity vector.

Definition 3 The tangent line to a smooth curve [公式] at the point [公式] is the straight line through the point [公式] in the direction of the velocity vector [公式] .

Tangent line and osculating plane of a curve

We let [公式] denote the length of a chord of a curve joining the points [公式] and [公式] and [公式] denote the length of a perpendicular dropped from [公式] onto the tangent line to [公式] at the point [公式] .

Lemma 1 Let [公式] be continuous in [公式] . Then,

[公式]

Proof: Trivial and left to the reader. [公式]

Theorem 1

[公式]

Proof: We have that [公式] and by Proposition 2 that

[公式]

We have, using properties of limits and keeping Lemma 1 in mind in the process,

[公式]

[公式]

Definition 4 A plane [公式] is called an osculating plane to a curve [公式] at a point [公式] if

[公式]

Theorem 2 At each point [公式] of a regular curve [公式] of class [公式] where [公式] , there is an osculating plane [公式] , and the vectors [公式] are orthogonal to its unit normal vector [公式] .

Proof: Based on the following diagram from [1],

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<p><small>Originally published at <a href="https://gmachine1729.wpcomstaging.com/2021/06/06/on-the-tangent-line-and-osculating-plane-of-a-curve/">狗和留美者不得入内</a>. You can comment here or <a href="https://gmachine1729.wpcomstaging.com/2021/06/06/on-the-tangent-line-and-osculating-plane-of-a-curve/#comments">there</a>.</small></p><div class="RichText ztext Post-RichText"> <p>Here, we will be working in <img src="https://www.zhihu.com/equation?tex=%5Cmathbb%7BR%7D%5E3" alt="[公式]" data-formula="\mathbb{R}^3" /> .</p> <h3>Analytic geometry prerequisites</h3> <p><b>Proposition 1</b> The distance between a point <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> and the plane given by <img src="https://www.zhihu.com/equation?tex=ax%2Bby%2Bcz+%3D+d" alt="[公式]" data-formula="ax+by+cz = d" /> is <img src="https://www.zhihu.com/equation?tex=%5Cfrac%7Bd-%28ax_1%2Bby_1%2Bcz_1%29%7D%7B%5Csqrt%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D%7D" alt="[公式]" data-formula="\frac{d-(ax_1+by_1+cz_1)}{\sqrt{a^2+b^2+c^2}}" /> .</p> <p><i>Proof</i>: A normal vector of the plane is <img src="https://www.zhihu.com/equation?tex=%28a%2Cb%2Cc%29" alt="[公式]" data-formula="(a,b,c)" /> . We plug in <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29+%2B+t%28a%2Cb%2Cc%29" alt="[公式]" data-formula="(x_1,y_1,z_1) + t(a,b,c)" /> to get</p> <p><img src="https://www.zhihu.com/equation?tex=a%28x_1%2Bat%29%2Bb%28y_1%2Bbt%29%2Bc%28z_1%2Bct%29+%3D+d%2C%5C%5C" alt="[公式]" data-formula="a(x_1+at)+b(y_1+bt)+c(z_1+ct) = d,\\" /> the solution of which is <img src="https://www.zhihu.com/equation?tex=t+%3D+%5Cfrac%7Bd-%28ax_1%2Bby_1%2Bcz_1%29%7D%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D" alt="[公式]" data-formula="t = \frac{d-(ax_1+by_1+cz_1)}{a^2+b^2+c^2}" /> . Since every unit of <img src="https://www.zhihu.com/equation?tex=t" alt="[公式]" data-formula="t" /> corresponds to <img src="https://www.zhihu.com/equation?tex=%5Csqrt%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D" alt="[公式]" data-formula="\sqrt{a^2+b^2+c^2}" /> of distance, we have for our answer<img src="https://www.zhihu.com/equation?tex=%5Cfrac%7Bd-%28ax_1%2Bby_1%2Bcz_1%29%7D%7B%5Csqrt%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D%7D" alt="[公式]" data-formula="\frac{d-(ax_1+by_1+cz_1)}{\sqrt{a^2+b^2+c^2}}" />. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>Proposition 2</b> The distance between a point <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> and a straight line given by <img src="https://www.zhihu.com/equation?tex=%28x_0%2Bat%2C+y_0%2Bbt%2C+z_0%2Bct%29" alt="[公式]" data-formula="(x_0+at, y_0+bt, z_0+ct)" /> can be obtained by the magnitude of a cross product.</p> <p><i>Proof</i>: As for this distance, it is obtained by taking the perpendicular with respect the straight line that contains <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> , which we shall call <img src="https://www.zhihu.com/equation?tex=h" alt="[公式]" data-formula="h" /> . We use <img src="https://www.zhihu.com/equation?tex=d" alt="[公式]" data-formula="d" /> to denote the distance between <img src="https://www.zhihu.com/equation?tex=%28x_0%2Cy_0%2Cz_0%29" alt="[公式]" data-formula="(x_0,y_0,z_0)" /> and <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> . One notices that <img src="https://www.zhihu.com/equation?tex=h" alt="[公式]" data-formula="h" /> is equal to <img src="https://www.zhihu.com/equation?tex=d+%5Csin+%5Ctheta" alt="[公式]" data-formula="d \sin \theta" /> , where <img src="https://www.zhihu.com/equation?tex=%5Ctheta" alt="[公式]" data-formula="\theta" /> is the angle between the straight line given in the proposition and the straight line connecting <img src="https://www.zhihu.com/equation?tex=%28x_0%2Cy_0%2Cz_0%29" alt="[公式]" data-formula="(x_0,y_0,z_0)" /> and <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> . We know that the magnitude of the cross product of two vectors is the product of their magnitudes and the sign of the angle between the two vectors, which completes our proof. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <h2>Preliminary definitions</h2> <p><b>Definition 1</b> A <i>regular curve</i> is a connected subset <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> of <img src="https://www.zhihu.com/equation?tex=%5Cmathbb%7BR%7D%5E3" alt="[公式]" data-formula="\mathbb{R}^3" /> homeomorphic to some <img src="https://www.zhihu.com/equation?tex=G" alt="[公式]" data-formula="G" /> that is a line segment <img src="https://www.zhihu.com/equation?tex=%5Ba%2Cb%5D" alt="[公式]" data-formula="[a,b]" /> or a circle of radius <img src="https://www.zhihu.com/equation?tex=1" alt="[公式]" data-formula="1" /> . If the homeomorphism <img src="https://www.zhihu.com/equation?tex=%5Cvarphi%3A+G+%5Cto+%5Cgamma" alt="[公式]" data-formula="\varphi: G \to \gamma" /> is in <img src="https://www.zhihu.com/equation?tex=C%5Ek" alt="[公式]" data-formula="C^k" /> for <img src="https://www.zhihu.com/equation?tex=k+%5Cgeq1" alt="[公式]" data-formula="k \geq1" /> and the rank of <img src="https://www.zhihu.com/equation?tex=%5Cvarphi" alt="[公式]" data-formula="\varphi" /> is maximal (equal to 1), then we say this curve is <i>k-fold continuously differentiable</i>. For <img src="https://www.zhihu.com/equation?tex=k%3D1" alt="[公式]" data-formula="k=1" /> , we say that <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> is <i>smooth</i>.</p> <p><b>Definition 2</b> Let a smooth curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> be given by the parametric equations</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D+%3D+%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D%28t%29+%3D+x%28t%29%5Cmathbf%7B%5Coverrightarrow%7Bi%7D%7D%2By%28t%29%5Cmathbf%7B%5Coverrightarrow%7Bj%7D%7D%2Bz%28t%29%5Cmathbf%7B%5Coverrightarrow%7Bk%7D%7D.%5C%5C" alt="[公式]" data-formula="\mathbf{\overrightarrow{r}} = \mathbf{\overrightarrow{r}}(t) = x(t)\mathbf{\overrightarrow{i}}+y(t)\mathbf{\overrightarrow{j}}+z(t)\mathbf{\overrightarrow{k}}.\\" /> The <i>velocity vector</i> of <img src="https://www.zhihu.com/equation?tex=%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D%28t%29" alt="[公式]" data-formula="\mathbf{\overrightarrow{r}}(t)" /> at <img src="https://www.zhihu.com/equation?tex=t+%3D+t_0" alt="[公式]" data-formula="t = t_0" /> is the derivative</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D%27%28t_0%29+%3D+x%27%28t_0%29%5Cmathbf%7B%5Coverrightarrow%7Bi%7D%7D%2By%27%28t_0%29%5Cmathbf%7B%5Coverrightarrow%7Bj%7D%7D%2Bz%27%28t_0%29%5Cmathbf%7B%5Coverrightarrow%7Bk%7D%7D.%5C%5C" alt="[公式]" data-formula="\mathbf{\overrightarrow{r}}'(t_0) = x'(t_0)\mathbf{\overrightarrow{i}}+y'(t_0)\mathbf{\overrightarrow{j}}+z'(t_0)\mathbf{\overrightarrow{k}}.\\" />The <i>velocity vector field</i> is the vector function <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t)" /> . The <i>speed</i> of <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}(t)" /> at <img src="https://www.zhihu.com/equation?tex=t+%3D+t_0" alt="[公式]" data-formula="t = t_0" /> is the length <img src="https://www.zhihu.com/equation?tex=%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C" alt="[公式]" data-formula="|\overrightarrow{\mathbf{r}}'(t_0)|" /> of the velocity vector.</p> <p><b>Definition 3</b> The <i>tangent line</i> to a smooth curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> at the point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0)" /> is the straight line through the point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%5Cin+%5Cgamma" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0) \in \gamma" /> in the direction of the velocity vector <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0)" /> .</p> <h2>Tangent line and osculating plane of a curve</h2> <p>We let <img src="https://www.zhihu.com/equation?tex=d" alt="[公式]" data-formula="d" /> denote the length of a chord of a curve joining the points <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Cgamma%28t_0%29" alt="[公式]" data-formula="P = \gamma(t_0)" /> and <img src="https://www.zhihu.com/equation?tex=P_1+%3D+%5Cgamma%28t_1%29" alt="[公式]" data-formula="P_1 = \gamma(t_1)" /> and <img src="https://www.zhihu.com/equation?tex=h" alt="[公式]" data-formula="h" /> denote the length of a perpendicular dropped from <img src="https://www.zhihu.com/equation?tex=P_1" alt="[公式]" data-formula="P_1" /> onto the tangent line to <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> at the point <img src="https://www.zhihu.com/equation?tex=P" alt="[公式]" data-formula="P" /> .</p> <p><b>Lemma 1</b> Let <img src="https://www.zhihu.com/equation?tex=%28x%28t%29%2Cy%28t%29%2Cz%28t%29%29" alt="[公式]" data-formula="(x(t),y(t),z(t))" /> be continuous in <img src="https://www.zhihu.com/equation?tex=t" alt="[公式]" data-formula="t" /> . Then,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bt_1+%5Cto+t_0%7D+%5B%28x%28t_0%29%2Cy%28t_0%29%2Cz%28t_0%29%29+%5Ctimes+%28x%28t_1%29%2Cy%28t_1%29%2Cz%28t_1%29%29%5D+%3D+%5Coverrightarrow%7B%5Cmathbf%7B0%7D%7D.%5C%5C" alt="[公式]" data-formula="\lim_{t_1 \to t_0} [(x(t_0),y(t_0),z(t_0)) \times (x(t_1),y(t_1),z(t_1))] = \overrightarrow{\mathbf{0}}.\\" /></p> <p><i>Proof</i>: Trivial and left to the reader. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>Theorem 1</b></p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bd+%5Cto+0%7D+%5Cfrac%7Bh%7D%7Bd%7D+%3D+%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7Bh%7D%7Bd%7D+%3D+0.%5C%5C" alt="[公式]" data-formula="\lim_{d \to 0} \frac{h}{d} = \lim_{t_1 \to t_0} \frac{h}{d} = 0.\\" /></p> <p><i>Proof</i>: We have that <img src="https://www.zhihu.com/equation?tex=d+%3D+%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C" alt="[公式]" data-formula="d = |\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)|" /> and by Proposition 2 that</p> <p><img src="https://www.zhihu.com/equation?tex=h+%3D+%5Cleft%7C%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%5Ctimes+%5B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%5D%5Cright%7C.%5C%5C" alt="[公式]" data-formula="h = \left|\frac{\overrightarrow{\mathbf{r}}'(t_0)}{|\overrightarrow{\mathbf{r}}'(t_0)|}\times [\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)]\right|.\\" /></p> <p>We have, using properties of limits and keeping Lemma 1 in mind in the process,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%5Clim_%7Bd+%5Cto+0%7D+%5Cfrac%7Bh%7D%7Bd%7D%26%3D%26%5Clim_%7Bt_1%5Cto+t_0%7D%5Cleft%7C%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%5Ctimes+%5Cleft%5B%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%5Cright%5D%5Cright%7C%5C%5C+%26%3D%26%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%5Ctimes+%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7Bt_1+-+t_0%7D%7C%7D%7B+%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7C%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7Bt_1+-+t_0%7D%7C%7D%5C%5C+%26%3D%26+%5Cfrac%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%5E2%7D+%3D+0.+%5Cend%7Beqnarray%7D%5C%5C" alt="[公式]" data-formula="\begin{eqnarray} \lim_{d \to 0} \frac{h}{d}&amp;=&amp;\lim_{t_1\to t_0}\left|\frac{\overrightarrow{\mathbf{r}}'(t_0)}{|\overrightarrow{\mathbf{r}}'(t_0)|}\times \left[\frac{\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)}{|\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)|}\right]\right|\\ &amp;=&amp;\lim_{t_1 \to t_0} \frac{|\overrightarrow{\mathbf{r}}'(t_0)\times \frac{\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)}{t_1 - t_0}|}{ |\overrightarrow{\mathbf{r}}'(t_0)||\frac{\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)}{t_1 - t_0}|}\\ &amp;=&amp; \frac{|\overrightarrow{\mathbf{r}}'(t_0)\times \overrightarrow{\mathbf{r}}'(t_0)|}{|\overrightarrow{\mathbf{r}}'(t_0)|^2} = 0. \end{eqnarray}\\" /></p> <p><img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>Definition 4</b> A plane <img src="https://www.zhihu.com/equation?tex=%5Calpha" alt="[公式]" data-formula="\alpha" /> is called an <i>osculating plane</i> to a curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> at a point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0)" /> if</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bd%5Cto+0%7D+%5Cfrac%7Bh%7D%7Bd%5E2%7D+%3D+%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7Bh%7D%7Bd%5E2%7D+%3D+0.%5C%5C" alt="[公式]" data-formula="\lim_{d\to 0} \frac{h}{d^2} = \lim_{t_1 \to t_0} \frac{h}{d^2} = 0.\\" /></p> <p><b>Theorem 2</b> At each point <img src="https://www.zhihu.com/equation?tex=P%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P= \overrightarrow{\mathbf{r}}(t_0)" /> of a regular curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> of class <img src="https://www.zhihu.com/equation?tex=C%5Ek" alt="[公式]" data-formula="C^k" /> where <img src="https://www.zhihu.com/equation?tex=k+%5Cgeq+2" alt="[公式]" data-formula="k \geq 2" /> , there is an osculating plane <img src="https://www.zhihu.com/equation?tex=%5Calpha" alt="[公式]" data-formula="\alpha" /> , and the vectors <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%2C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0),\overrightarrow{\mathbf{r}}''(t_0)" /> are orthogonal to its unit normal vector <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}}" /> .</p> <p><i>Proof</i>: Based on the following diagram from [1],</p> <figure data-size="normal"><noscript><img src="https://i2.wp.com/pic4.zhimg.com/v2-d4b39ec764b969ae82b0c141ec3e117f_b.jpg?w=327&#038;ssl=1" data-caption="" data-size="normal" data-rawwidth="327" data-rawheight="234" class="content_image" data-recalc-dims="1" /></noscript><img class="content_image lazy" src="data:;base64,<svg xmlns='http://www.w3.org/2000/svg' width='327' height='234'></svg>&#8221; width=&#8221;327&#8243; data-caption=&#8221;&#8221; data-size=&#8221;normal&#8221; data-rawwidth=&#8221;327&#8243; data-rawheight=&#8221;234&#8243; data-actualsrc=&#8221;<a href="https://pic4.zhimg.com/v2-d4b39ec764b969ae82b0c141ec3e117f_b.jpg&#038;#8221" rel="nofollow">https://pic4.zhimg.com/v2-d4b39ec764b969ae82b0c141ec3e117f_b.jpg&#038;#8221</a>; /></figure> <p>we have</p> <p><img src="https://www.zhihu.com/equation?tex=d+%3D+%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_1%29+-+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%7C%2C+%5Cqquad+h+%3D+%7C%5Clangle+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_1%29+-+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle%7C.+%5Cqquad+%281%29%5C%5C" alt="[公式]" data-formula="d = |\overrightarrow{\mathbf{r}}(t_1) - \overrightarrow{\mathbf{r}}(t_0)|, \qquad h = |\langle \overrightarrow{\mathbf{r}}(t_1) - \overrightarrow{\mathbf{r}}(t_0), \overrightarrow{\boldsymbol{\beta}}\rangle|. \qquad (1)\\" /> We first prove the existence of osculating plane, for which there are two cases:</p> <ol> <li><img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29+%5Cneq+0" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0) \times \overrightarrow{\mathbf{r}}''(t_0) \neq 0" /> .</li> <li><img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29+%3D+0" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0) \times \overrightarrow{\mathbf{r}}''(t_0) = 0" /> .</li> </ol> <p>In the first case, we simply define the unit vector <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D+%3D+%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%7C%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}} = \frac{\overrightarrow{\mathbf{r}}(t_0) \times \overrightarrow{\mathbf{r}}(t_0)}{|\overrightarrow{\mathbf{r}}(t_0) \times \overrightarrow{\mathbf{r}}(t_0)|}" /> and in the second case, take any <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}}" /> orthogonal to <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0)" /> , which is non-zero by definition of regular curve. In both cases, we have</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clangle+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle+%3D+%5Clangle+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle+%3D+0.+%5Cqquad+%282%29%5C%5C" alt="[公式]" data-formula="\langle \overrightarrow{\mathbf{r}}'(t_0), \overrightarrow{\boldsymbol{\beta}}\rangle = \langle \overrightarrow{\mathbf{r}}''(t_0), \overrightarrow{\boldsymbol{\beta}}\rangle = 0. \qquad (2)\\" /> Let <img src="https://www.zhihu.com/equation?tex=%5Calpha" alt="[公式]" data-formula="\alpha" /> be the plane passing through the point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0)" /> and orthogonal to <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}}" /> . By Taylor&#8217;s formula,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_1%29+-+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%28t_1+-+t_0%29+%2B+%5Cfrac%7B1%7D%7B2%7D%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29%28t_1+-+t_0%29%5E2+%2B+%5Coverrightarrow%7B%5Cboldsymbol%7Bo%7D%7D%28%7Ct_1+-+t_0%7C%5E2%29.+%5Cqquad+%283%29%5C%5C" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}(t_1) - \overrightarrow{\mathbf{r}}(t_0) = \overrightarrow{\mathbf{r}}'(t_0)(t_1 - t_0) + \frac{1}{2}\overrightarrow{\mathbf{r}}''(t_0)(t_1 - t_0)^2 + \overrightarrow{\boldsymbol{o}}(|t_1 - t_0|^2). \qquad (3)\\" /> Applying <img src="https://www.zhihu.com/equation?tex=%282%29%2C%283%29" alt="[公式]" data-formula="(2),(3)" /> to <img src="https://www.zhihu.com/equation?tex=%281%29" alt="[公式]" data-formula="(1)" /> gives</p> <p><img src="https://www.zhihu.com/equation?tex=h+%3D+%7C%5Clangle+%5Coverrightarrow%7B%5Cboldsymbol%7Bo%7D%7D%28%7Ct_1+-+t_0%7C%5E2%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle%7C%2C+%5Cqquad+d+%3D%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%28t_1-t_0%29%2B%5Coverrightarrow%7B%5Cboldsymbol%7Bo%7D%7D%28%7Ct_1-t_0%7C%29%7C.%5C%5C" alt="[公式]" data-formula="h = |\langle \overrightarrow{\boldsymbol{o}}(|t_1 - t_0|^2), \overrightarrow{\boldsymbol{\beta}}\rangle|, \qquad d =|\overrightarrow{\mathbf{r}}'(t_0)(t_1-t_0)+\overrightarrow{\boldsymbol{o}}(|t_1-t_0|)|.\\" /> From this, one can verify without difficulty that</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7Bh%7D%7Bd%5E2%7D+%3D+0.%5C%5C" alt="[公式]" data-formula="\lim_{t_1 \to t_0} \frac{h}{d^2} = 0.\\" /> For the other part of this theorem, we simply uses <img src="https://www.zhihu.com/equation?tex=%281%29" alt="[公式]" data-formula="(1)" /> and <img src="https://www.zhihu.com/equation?tex=%283%29" alt="[公式]" data-formula="(3)" /> on the limit equal to zero to deduce the desired orthogonality relations. The details will be left to the reader as an exercise. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>References</b></p> <ul> <li>[1] <a class=" wrap external" href="https://link.zhihu.com/?target=https%3A//gmachine1729.wpcomstaging.com/%25e5%2590%258d%25e5%258d%2595-c%25d0%25bf%25d0%25b8%25d1%2581%25d0%25ba%25d0%25b8-lists/%25e5%2590%258d%25e5%258d%2595-c%25d0%25bf%25d0%25b8%25d1%2581%25d0%25ba%25d0%25b8-lists-%25e6%2595%25b0%25e5%25ad%25a6%25e5%2592%258c%25e7%2589%25a9%25e7%2590%2586%25e4%25b9%25a6-math-and-physics-books/victor-andreevich-toponogov-vladimir-rovenski-differential-geometry-of-curves-and-surfaces_-a-concise-guide-birkha%25cc%2588user-boston-2005/" target="_blank" rel="noopener noreferrer">Victor Andreevich Toponogov, Vladimir Rovenski &#8211; Differential Geometry of Curves and Surfaces: A Concise Guide (2005)</a></li> </ul> </div>

On the chain rule and change of variables of integrals

Originally published at 狗和留美者不得入内. You can comment here or there.

Theorem 1 (Chain rule) Let [公式] , [公式] , where [公式] and [公式] are open in [公式] , such that [公式] are differentiable on their respective domains. Then [公式] is also differentiable on [公式] , with [公式] for all [公式] .

Proof: We first assume that there exists a neighborhood [公式] of [公式] for which [公式] . This happens in the case of [公式] by inverse function theorem. In that case, by the definition of derivative and its properties, we have

[公式] In the case of [公式] , we have that for all [公式] ,

[公式]

From this, we easily verifies that [公式] , which means that [公式] is differentiable at [公式] and in the case of [公式] , [公式] must hold as well. [公式]

Lemma 1 Let [公式] , [公式] be differentiable [公式] with [公式] and [公式] . Then,

[公式]

Instead of [公式] , one can also use any closed interval of [公式] .

Proof: Follows directly from Fundamental Theorem of Calculus. See Theorem 2 (Newton-Leibniz axiom) of [1]. [公式]

Lemma 1 is a statement of invariance of integral along parameterized smooth paths with the same endpoints.

Theorem 2 (Change of variables or u-substitution in integration) Let [公式] be any differentiable function of [公式] on [公式] , which is continuous on [公式] , and [公式] be Riemann integrable on intervals in its domain. Then,

[公式]

Proof: Let [公式] be an antiderivative of [公式] . By the Fundamental Theorem of Calculus, it suffices to show that the left hand side of [公式] is equal to [公式] , which can be done by applying Lemma 1 accordingly. [公式]

Theorem 3 (Integration by parts) Let [公式] be differentiable functions on [公式] and continuous on [公式] . Then,

[公式]

Proof: We have

[公式]

Rearranging the above completes the proof. [公式]

References