Sheng Li (gmachine1729) wrote,
Sheng Li
gmachine1729

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.
Tags: uncategorized
Subscribe

  • Post a new comment

    Error

    Anonymous comments are disabled in this journal

    default userpic

    Your reply will be screened

  • 0 comments