Riesz's lemma

Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense. The lemma may also be called the Riesz lemma or Riesz inequality. It can be seen as a substitute for orthogonality when the normed space is not an inner product space. If is a reflexive Banach space then this conclusion is also true when Proof The proof can be found in functional analysis texts such as Kreyszig. An online proof from Prof. Paul Garrett is available. Metric reformulation As usual, let denote the canonical metric induced by the norm, call the set of all vectors that are a distance of from the origin , and denote the distance from a point to the set by The inequality holds if and only if for all and it formally expresses the notion that distance between and is at least Because every vector subspace (such as ) contains the origin substituting in this infimum shows that for every vector In particular, when is a unit vector. Using this new notation, the conclusion of Riesz's lemma may be restated more succinctly as: holds for some Using this new terminology, Riesz's lemma may also be restated in plain English as: Given any closed proper vector subspace of a normed space for any desired minimum distance less than there exists some vector in the unit sphere of that is this desired distance away from the subspace. Minimum distances not satisfying the hypotheses When is trivial then it has no vector subspace and so Riesz's lemma holds vacuously for all real numbers The remainder of this section will assume that which guarantees that a unit vector exists. The inclusion of the hypotheses can be explained by considering the three cases: is non-negative, and The lemma holds when since every unit vector satisfies the conclusion The hypotheses is included solely to exclude this trivial case and is sometimes omitted from the lemma's statement. Riesz's lemma is always false when because for every unit vector the required inequality fails to hold for (since ).