corollary of Hahn-Banach

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Matrix Analysis

Corollary of Hahn-Banach

Let $V, | | \cdot | |$ be a NLS over $K$ . If $x \in V$ , then there exists $f \in V^{*}$ such that $| | f | |_{V^{*}} = 1$ and $f (x) = | | x | |_{V}$

(without proof)

Functional Analysis

Theorem

if $V$ is a normed space, then for all $v \in V ∖ {0}$ , there exists $f \in V^{'}$ such that

\begin{aligned} | | f | | & = 1 \\ f (v) & = | | v | | \end{aligned}

Proof

Define $u : C v \to C$ as $u (λ v) = λ | | v | |$ . Then $| u (t) | \leq | | t | |$ for all $t \in C v$ and also $u (v) = | | v | |$ . Thus, by the Hahn-Banach theorem, there exists some $f \in V^{'}$ that extends $u$ , ie $f (v) = u (v) = | | v | |$ . But then

Then
$| | f (t) | | \leq | | t | |$ for all $t \in V$ , ie

1 = f (\frac{v}{| | v | |}) \leq | | f | | ⟹ | | f | | = 1

Thus we have found a linear functional as desired.

File	Last Modified
Functional Analysis Lecture 6	2025-07-08
the functional to the double dual is isometric	2025-06-12

corollary of Hahn-Banach

Matrix Analysis

Functional Analysis

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