It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense.
be a closed proper vector subspace of a normed space
is a reflexive Banach space then this conclusion is also true when
denote the canonical metric induced by the norm, call the set
from the origin the unit sphere, and denote the distance from a point
Using this new notation, the conclusion of Riesz's lemma may be restated more succinctly as:
Using this new terminology, Riesz's lemma may also be restated in plain English as: The proof[3] can be found in functional analysis texts such as Kreyszig.
and so Riesz's lemma holds vacuously for all real numbers
is included solely to exclude this trivial case and is sometimes omitted from the lemma's statement.
for consideration, in which case the statement of Riesz’s lemma becomes: When
is reflexive if and only if for every closed proper vector subspace
of all bounded sequences, Riesz’s lemma does not hold for
[5] However, every finite dimensional normed space is a reflexive Banach space, so Riesz’s lemma does holds for
when the normed space is finite-dimensional, as will now be shown.
is continuous, its image on the closed unit ball
must be a compact subset of the real line, proving the claim.
or stated in plain English, these vectors are all separated from each other by a distance of more than
Such an infinite sequence of vectors cannot be found in the unit sphere of any finite dimensional normed space (just consider for example the unit circle in
contains no convergent subsequence, which implies that the closed unit ball is not compact.
Riesz's lemma can be applied directly to show that the unit ball of an infinite-dimensional normed space
This can be used to characterize finite dimensional normed spaces: if
is finite dimensional if and only if the closed unit ball in
More generally, if a topological vector space
Namely, if a topological vector space is finite dimensional, it is locally compact.
[6] Therefore local compactness characterizes finite-dimensionality.
This classical result is also attributed to Riesz.
[7] The spectral properties of compact operators acting on a Banach space are similar to those of matrices.
Riesz's lemma is essential in establishing this fact.
As detailed in the article on infinite-dimensional Lebesgue measure, 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