# 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 of any vector space shall be either or . Moreover, subspace will always denote the subspace of a vector space.

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

1. Absolute homogeneity: for all , , .
2. Triangle inequality: for all , .

Proposition 1 For any seminorm , .

Proof: Follows directly from absolute homogeneity.

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

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

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

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

Definition 6 A series converges absolutely iff .

Theorem 1 A normed vector space 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 , for some . Now take any such that , which of course means that . To show that it converges, it suffices to show that 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>
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• #### Some thoughts on Greek, Persian, Arab, Indian, and Chinese science

Originally published at 狗和留美者不得入内. You can comment here or there. Reading more about history, I’ve come to notice that China’s main…

• #### Why the 1700 Japanese could have discovered calculus and modern science by 2200 or 2700

Originally published at 狗和留美者不得入内. You can comment here or there. Some thoughts on Greek, Persian, Arab, Indian, and Chinese science In the…

• #### A History of Nobel Physicists from Wartime Japan

转自 https://www.scientificamerican.com/article/physicists-in-wartime-japan/ Nobel laureate Yoichiro Nambu co-authored this piece about the most…

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