Riesz's lemma [SEP] named after
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 linear 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 one is not in an inner product space.
The result.
Riesz's Lemma. Let "X" be a normed space, "Y" be a closed proper subspace of
acting on a Banach space are similar to those of matrices. Riesz's lemma is essential in establishing this fact.
Riesz's lemma guarantees that any infinite-dimensional normed space contains a sequence of unit vectors {"x"} with formula_5 for 0 < "α" < 1. This is useful in showing the non-existence of certain measures on infinite-dimensional Banach spaces. Riesz's lemma also shows that the identity operator on a Banach space "X" is compact if and only if "X"