Riesz's lemma

Summary

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 vector 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 the normed space is not an inner product space.

Statement edit

Riesz's lemma[1] — Let   be a closed proper vector subspace of a normed space   and let   be any real number satisfying   Then there exists a vector   in   of unit norm   such that   for all   in  

If   is a reflexive Banach space then this conclusion is also true when  [2]

Metric reformulation

As usual, let   denote the canonical metric induced by the norm, call the set   of all vectors that are a distance of   from the origin the unit sphere, and denote the distance from a point   to the set   by

 
The inequality   holds if and only if   for all   and it formally expresses the notion that the distance between   and   is at least   Because every vector subspace (such as  ) contains the origin   substituting   in this infimum shows that   for every vector   In particular,   when   is a unit vector.

Using this new notation, the conclusion of Riesz's lemma may be restated more succinctly as:   holds for some  

Using this new terminology, Riesz's lemma may also be restated in plain English as:

Given any closed proper vector subspace of a normed space   for any desired minimum distance   less than   there exists some vector in the unit sphere of   that is at least this desired distance away from the subspace.

The proof[3] can be found in functional analysis texts such as Kreyszig.[4] An online proof from Prof. Paul Garrett is available.

Proof

Consider any   and denote its distance from   by  . Clearly,   since   is closed. Take any  . By the definition of an infimum, there is a   such that

 

(1)

(note that   since  ). Let

 

Then  , and we show that   for every  . We have

 

where

 

The form of   shows that  . Hence  , by the definition of  . Writing   out and using (1), we obtain

 

Since   was arbitrary, this completes the proof.

Minimum distances   not satisfying the hypotheses

When   is trivial then it has no proper vector subspace   and so Riesz's lemma holds vacuously for all real numbers   The remainder of this section will assume that   which guarantees that a unit vector exists.

The inclusion of the hypotheses   can be explained by considering the three cases:  ,   and   The lemma holds when   since every unit vector   satisfies the conclusion   The hypotheses   is included solely to exclude this trivial case and is sometimes omitted from the lemma's statement.

Riesz's lemma is always false when   because for every unit vector   the required inequality   fails to hold for   (since  ). Another of consequence of   being impossible is that the inequality   holds if and only if equality   holds.

Reflexivity edit

This leaves only the case   for consideration, in which case the statement of Riesz's lemma becomes:

for every closed proper vector subspace   of   there exists some vector   of unit norm that satisfies  

When   is a Banach space, then this statement is true if and only if   is a reflexive space.[2] Explicitly, a Banach space   is reflexive if and only if for every closed proper vector subspace   there is some vector   on the unit sphere of   that is always at least a distance of   away from the subspace.

For example, if the reflexive Banach space   is endowed with the usual   Euclidean norm and if   is the   plane then the points   satisfy the conclusion   If   is  -axis then every point   belonging to the unit circle in the   plane satisfies the conclusion   But if   was endowed with the   taxicab norm (instead of the Euclidean norm), then the conclusion   would be satisfied by every point   belonging to the "diamond"   in the   plane (a square with vertices at   and  )

In a non-reflexive Banach space, such as the Lebesgue space   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. When the dimension of   is finite then the closed unit ball   is compact. Since the distance function   is continuous, its image on the closed unit ball   must be a compact subset of the real line, proving the claim.

Some consequences edit

Riesz's lemma guarantees that for any given   every infinite-dimensional normed space contains a sequence   of (distinct) unit vectors satisfying   for   or stated in plain English, these vectors are all separated from each other by a distance of more than   while simultaneously also all lying on the unit sphere. 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  ).

This sequence can be constructed by induction for any constant   Start by picking any element   from the unit sphere. Let   be the linear span of   and (using Riesz's lemma) pick   from the unit sphere such that

  where  

This sequence   contains no convergent subsequence, which implies that the closed unit ball is not compact.

Characterization of finite dimension edit

Riesz's lemma can be applied directly to show that the unit ball of an infinite-dimensional normed space   is never compact. This can be used to characterize finite dimensional normed spaces: if   is a normed vector space, then   is finite dimensional if and only if the closed unit ball in   is compact.

More generally, if a topological vector space   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.[6] Therefore local compactness characterizes finite-dimensionality. This classical result is also attributed to Riesz. A short proof can be sketched as follows: let   be a compact neighborhood of the origin in   By compactness, there are   such that

 

We claim that the finite dimensional subspace   spanned by   is dense in   or equivalently, its closure is   Since   is the union of scalar multiples of   it is sufficient to show that   By induction, for every  

 
But compact sets are bounded, so   lies in the closure of   This proves the result. For a different proof based on Hahn–Banach theorem see Crespín (1994).[7]

Spectral theory 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.

Other applications edit

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   is compact if and only if   is finite-dimensional.[8]

See also edit

References edit

  1. ^ Rynne & Youngson 2008, p. 47.
  2. ^ a b Diestel 1984, p. 6.
  3. ^ "Riesz's lemma". PlanetMath.
  4. ^ Kreyszig 1978.
  5. ^ "An example where the supremum of Riesz's Lemma is not achieved".
  6. ^ Tao, Terence (24 May 2011). "Locally compact topological vector spaces".
  7. ^ Crespín, Daniel (1994). "Hahn–Banach theorem implies Riesz theorem" (PDF). Portugaliae Mathematica. 51 (2): 217–218. MR 1277990.
  8. ^ Kreyszig (1978, Theorem 2.5-3, 2.5-5)
  • Diestel, Joe (1984). Sequences and series in Banach spaces. New York: Springer-Verlag. ISBN 0-387-90859-5. OCLC 9556781.
  • Hashimoto, Kazuo; Nakamura, Gen; Oharu, Shinnosuke (1986-01-01). "Riesz's lemma and orthogonality in normed spaces" (PDF). Hiroshima Mathematical Journal. 16 (2). Hiroshima University - Department of Mathematics. doi:10.32917/hmj/1206130429. ISSN 0018-2079.
  • Kreyszig, Erwin (1978). Introductory functional analysis with applications. New York: John Wiley & Sons. ISBN 0-471-50731-8. OCLC 2818701.
  • Riesz, Frederic; Sz.-Nagy, Béla (1990) [1955]. Functional Analysis. Translated by Boron, Leo F. New York: Dover Publications. ISBN 0-486-66289-6. OCLC 21228994.
  • Rynne, Bryan P.; Youngson, Martin A. (2008). Linear Functional Analysis (2nd ed.). London: Springer. ISBN 978-1848000049. OCLC 233972987.