Lagmental Vicfred

Hahn--Banach Extends a Bounded Functional without Growing Its Norm by Vicfred

tags: mathematics, functional analysis, Banach spaces, operator theory

A continuous linear functional on a subspace extends to the entire normed space with the same norm. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Start locally

A Banach space \(X\) is complete in its norm, and a bounded linear operator \(T:X\to Y\) has norm \(\|T\|=\sup_{\|x\|\le1}\|Tx\|\). Completeness powers the major structural theorems.

$$ M\le X,\qquad f\in M^\ast $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \exists F\in X^\ast,\qquad F|_M=f,\quad\|F\|=\|f\| $$

Compute before generalising

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ x_0\notin\overline M\Longrightarrow\exists F\in X^\ast:\quad F|_M=0,\quad F(x_0)=1,\quad\|F\|=\frac1{\operatorname{dist}(x_0,M)} $$

The global view

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad M\le X,\qquad f\in M^\ast,\\[5pt] \mathsf{C}\;&:\quad \exists F\in X^\ast,\qquad F|_M=f,\quad\|F\|=\|f\|. \end{aligned} $$

Edge conditions

Finite-dimensional intuition can fail badly in infinite dimensions. Closed, bounded sets need not be compact, and linear maps need not be bounded automatically.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \exists F\in X^\ast,\qquad F|_M=f,\quad\|F\|=\|f\| \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Tue 04 September 2018. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.