Lagmental Vicfred

Uniform Boundedness Turns Pointwise Bounds into an Operator-Norm Bound by Vicfred

Last updated: Fri 28 February 2020

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

A pointwise bounded family of operators on a Banach space is uniformly bounded in norm. This is a compact note, but the quantifiers and hypotheses stay on the page.

Objects and notation

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.

$$ \mathcal T\subseteq\mathcal B(X,Y),\qquad\sup_{T\in\mathcal T}\|Tx\|<\infty\ \forall x\in X $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ \sup_{T\in\mathcal T}\|T\|<\infty $$

Push the symbols

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ E_n=\{x:\sup_{T\in\mathcal T}\|Tx\|\le n\},\qquad X=\bigcup_{n\ge1}E_n\quad\overset{\text{Baire}}{\Longrightarrow}\quad\operatorname{int}E_N\ne\varnothing $$

Structural reading

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad \mathcal T\subseteq\mathcal B(X,Y),\qquad\sup_{T\in\mathcal T}\|Tx\|<\infty\ \forall x\in X,\\[5pt] \mathsf{C}\;&:\quad \sup_{T\in\mathcal T}\|T\|<\infty. \end{aligned} $$

A hypothesis worth keeping

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \sup_{T\in\mathcal T}\|T\|<\infty \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Sat 10 August 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.