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

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  • $\begingroup$ 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? $\endgroup$ – fuglede Aug 28 '16 at 15:05
  • $\begingroup$ Yes, it would suffice. $\endgroup$ – Konstantin Aug 28 '16 at 16:09
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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.

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    $\begingroup$ Wonderful explanation, thank you! $\endgroup$ – Konstantin Aug 28 '16 at 18:34

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