Lemma from Riesz

The Riesz's lemma , named after the Hungarian mathematician Frigyes Riesz , is a set of functional analysis on completed sub-spaces of normed spaces .

statement

A normalized space , a closed true subspace of and a real number are given . ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle 0<\theta <1}$

Then there exists an element with such that: ${\displaystyle {\hat {x}}\in X}$ ${\displaystyle \|{\hat {x}}\|=1}$

${\displaystyle \mathrm {d} ({\hat {x}},U)=\inf _{u\in U}\|{\hat {x}}-u\|\geq \theta }$.

If finite dimensional or more generally reflexive , then a choice can be made. ${\displaystyle U}$ ${\displaystyle \theta =1}$

motivation

In a finite-dimensional Euclidean space there is a unit vector perpendicular to it for every real subspace . The distance of any point from to is then at least one, the value one is assumed to be exactly . ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle u}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle u=0}$

In a standardized space, the term “standing vertically” is generally not definable. In this respect, the formulation of Riesz's lemma is a useful generalization. It is also not a matter of course that vectors with a positive distance to it still exist outside a subspace.

Evidence sketch

There is a point outside the real subspace . Since is complete, the distance from must be positive. Be a given and a point in with ${\displaystyle w}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle w}$ ${\displaystyle U}$ ${\displaystyle 0<\theta <1}$ ${\displaystyle u_{0}}$ ${\displaystyle U}$

${\displaystyle \|w-u_{0}\|\leq {\frac {1}{\theta }}\mathrm {d} (w,U)}$.

Choose as element : ${\displaystyle u_{0}\in U}$

${\displaystyle {\hat {x}}:={\frac {w-u_{0}}{\|w-u_{0}\|}}}$

This is standardized by construction. The following applies to any one : ${\displaystyle u\in U}$

${\displaystyle \|{\hat {x}}-u\|=\left\|{\frac {w-u_{0}}{\|w-u_{0}\|}}-u\right\|={\frac {1}{\|w-u_{0}\|}}\cdot \|w-\underbrace {u_{0}-\|w-u_{0}\|\cdot u} _{\in U}\|\geq {\frac {\mathrm {d} (w,U)}{\|w-u_{0}\|}}\geq \theta }$.

The following applies to the distance:

${\displaystyle \mathrm {d} ({\hat {x}},U)\geq {\frac {\mathrm {d} (w,U)}{\|w-u_{0}\|}}\geq \theta }$.

Inferences

From Riesz's lemma it follows that every normed space in which the closed unit sphere is compact must be finite-dimensional. The reverse of this theorem is also correct ( Riesz compactness theorem ).

Individual evidence

1. ↑ Harro Heuser: functional analysis , Teubner-Verlag (1975), ISBN 3-519-02206-0 , auxiliary set 10.2
2. ^ Joseph Diestel: Sequences and Series in Banach Spaces , Springer-Verlag (1984), ISBN 3-540-90859-5 , chap. I, lemma on page 2
3. ↑ Dirk Werner : Functional Analysis. 6th, corrected edition, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , p. 27.