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.

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

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.

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.

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

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

More generally, if a topological vector space X is locally compact, then it is finite dimensional. The converse of this is also true. Namely, if a topological vector space is finite dimensional, it is locally compact. 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

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