Lagmental Vicfred

Riesz Representation Identifies Hilbert-Space Functionals by Vicfred

Last updated: Sat 24 June 2023

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

Every continuous linear functional on a Hilbert space is inner product with one unique vector. I will separate the object being defined from the consequence being claimed.

Definitions first

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.

$$ H\ \text{Hilbert},\qquad\ell\in H^\ast $$

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.

$$ \exists!\,y\in H,\qquad\ell(x)=\langle x,y\rangle,\quad\|\ell\|=\|y\| $$

A small case in full

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ H=\ker\ell\oplus(\ker\ell)^\perp,\qquad y=\frac{\overline{\ell(z)}}{\|z\|^2}z\quad(0\ne z\in(\ker\ell)^\perp) $$

The reusable statement

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad H\ \text{Hilbert},\qquad\ell\in H^\ast,\\[5pt] \mathsf{C}\;&:\quad \exists!\,y\in H,\qquad\ell(x)=\langle x,y\rangle,\quad\|\ell\|=\|y\|. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \exists!\,y\in H,\qquad\ell(x)=\langle x,y\rangle,\quad\|\ell\|=\|y\| \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Mon 01 November 2021. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.