Riesz's lemma

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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[edit]

Riesz's Lemma. Let X be a normed 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(xY) = 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[edit]

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 xn from the unit sphere such that

for a constant 0 < α < 1, where Yn−1 is the linear span of {x1 ... xn−1} and .

Clearly {xn} 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[2]. 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, ..., cnC such that

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

for every m. But compact sets are bounded, so C lies in the closure of Y. This proves the result. For a different proof based on Hahn-Banach Theorem see [3].

Some consequences[edit]

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 {xn} with 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 is finite-dimensional.[4]

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).

References[edit]

  1. ^ Rynne, Bryan P.; Youngson, Martin A. (2008). Linear Functional Analysis (2nd ed.). London: Springer. p. 47. ISBN 978-1848000049.
  2. ^ https://terrytao.wordpress.com/2011/05/24/locally-compact-topological-vector-spaces/
  3. ^ https://www.emis.de/journals/PM/51f2/pm51f205.pdf/
  4. ^ Kreyszig (1978, Theorem 2.5-3, 2.5-5)
  • Kreyszig, Erwin (1978), Introductory functional analysis with applications, John Wiley & Sons, ISBN 0-471-50731-8