Graphical explanation of Riesz's lemma

Question

Does there exist an intuitive graphical explanation of Riesz's lemma?

How much do you demand of such an explanation? Would it suffice to illustrate it for two-dimensional subspaces of $\mathbb{R}^3$, or is it too intuitive in that case? — fuglede, Aug 28 '16 at 15:05

fuglede · Accepted Answer · 2016-09-04T08:29:19.837

It's a bit hard to say beforehand if this is useful or not, but here are some pictures; we use the notation from Wikipedia:

Riesz's Lemma. Let $\color{black}X$ be a normed linear space, $\color{green}Y$ be a closed proper subspace of $\color{black}X$ and $\color{\lightblue}\alpha$ be a real number with $0 < \color{\lightblue}\alpha\color{black} < 1$. Then there exists an $\color{orange}x$ in $X$ with $\color{red}{|x| = 1}$ such that $\lvert \color{\orange} x − y\rvert > \color{\lightblue}α$ for all y in $\color{green}Y$.

We picture an $\color{lightblue}\alpha$-neighbourhood around $\color{green}Y$. Then the claim of the theorem is that at least one point in the unit sphere, $\color{red}{\{x \mid \lvert x \rvert = 1\}}$, is outside of that neighbourhood.

$$X = \mathbb{R}^2$ and $Y$ is one-dimensional.$ $X = \mathbb{R}^2$ and $Y$ is one-dimensional.

$$X = \mathbb{R}^3$ and $Y$ is one-dimensional.$ $X = \mathbb{R}^3$ and $Y$ is one-dimensional.

$$X = \mathbb{R}^3$ and $Y$ is two-dimensional.$ $X = \mathbb{R}^3$ and $Y$ is two-dimensional.

Wonderful explanation, thank you! – Konstantin Aug 28 '16 at 18:34 — Konstantin, Aug 28 '16 at 18:34

Graphical explanation of Riesz's lemma

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