June 10th, 2021

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>

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.