Riesz's lemma

The lemma of Riesz, named after the Hungarian mathematician Frigyes Riesz, is a set of functional analysis on closed subspaces of normed spaces.

Statement

Given a normed space, a closed real subspace of and a real number. Then an element exists with, so that applies:

Is finite, then one can be selected. ( It is enough to assume that is reflexive. )

Motivation

In a finite-dimensional Euclidean space, for every real subspace U that is perpendicular to a unit vector x. The distance of any point u of U then x is at least one, the value one is exactly assumed for u = 0.

In a normed space is the concept of "vertical standing " in general not definable. In this respect, the formulation of the lemma of Riesz is a useful generalization. Also, it is not self-evident that outside a subspace vectors with positive distance still exist to this.

Sketch of proof

There is a point w outside the proper subset U. Since U is complete, the distance of w must be positive to U. Be a preset ( for the statement is trivial) and v be a point in U with

Define, that is, x is a vector with norm 1 for the distance from x to U we have

Conclusions

It follows from the lemma of Riesz that every normed space in which the closed unit ball is compact, must be finite -dimensional. The converse of this theorem is correct ( compactness theorem of Riesz ).