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

Before stating the result, we fix some notation. Let X be a normed linear space with norm |·| and x be an element of X. Let Y be a closed subspace in X. The distance between an element x and Y is defined by

$d(x, Y) = \inf_{y \in Y} |x - y|.$

Now we can state the Lemma:

Riesz's Lemma. Let X be a normed linear space, Y be a closed proper subspace of X and α be a real number with 0 < α < 1. Then there exists an x in X with |x| = 1 such that |x − y| > α for all y in Y.^[1]

Remark 1. For the finite-dimensional case, equality can be achieved. In other words, there exists x of unit norm such that d(x, Y) = 1. When dimension of X is finite, the unit ball B ⊂ X is compact. Also, the distance function d(· , Y) is continuous. Therefore its image on the unit ball B must be a compact subset of the real line, proving the claim.

Remark 2. The space ℓ_∞ of all bounded sequences shows that the lemma does not hold for α = 1.

The proof can be found in functional analysis texts such as Kreyszig. An online proof from Prof. Paul Garrett is available.

Converse

Riesz's lemma can be applied directly to show that the unit ball of an infinite-dimensional normed space X is never compact: Take an element x₁ from the unit sphere. Pick x_n from the unit sphere such that

$d(x_n, Y_{n-1}) > \alpha$ for a constant 0 < α < 1, where Y_n−1 is the linear span of {x₁ ... x_n−1}.

Clearly {x_n} contains no convergent subsequence and the noncompactness of the unit ball follows.

The converse of this, in a more general setting, is also true. If a topological vector space X is locally compact, then it is finite dimensional. Therefore local compactness characterizes finite-dimensionality. This classical result is also attributed to Riesz. A short proof can be sketched as follows: let C be a compact neighborhood of 0 ∈ X. By compactness, there are c₁, ..., c_n ∈ C such that

$C = \bigcup_{i=1}^n \; \left( c_i + \frac{1}{2} C \right).$

We claim that the finite dimensional subspace Y spanned by {c_i}, or equivalently, its closure, is X. Since scalar multiplication is continuous, it's enough to show C ⊂ Y. Now, by induction,

$C \sub Y + \frac{1}{2^m} C$

for every m. But compact sets are bounded, so C lies in the closure of Y. This proves the result.

Some consequences

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.

Riesz's lemma guarantees that any infinite-dimensional normed space contains a sequence of unit vectors {x_n} with $|x_n - x_m| > \alpha$ for 0 < α < 1. This is useful in showing the non-existence of certain measures on infinite-dimensional Banach spaces.

One can also use this lemma to demonstrate whether or not the normed vector space X is finite dimensional or otherwise: if the closed unit ball is compact, then X is finite dimensional (we proceed by contradiction to this proof this results).

Naming

Some authors, such as Reed and Simon, use the name "Riesz lemma" to refer to the Riesz representation theorem, which is unrelated to the Riesz lemma described in this article.

References

1. Rynne, Bryan P.; Youngson, Martin A. (2008). Linear Functional Analysis (2nd ed.). London: Springer. p. 47. ISBN 978-1848000049.