# Riesz's lemma

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**Riesz' lemma** (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions which guarantee that a subspace in a normed linear space is dense.

## 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

Now we can state the Lemma:

Riesz's Lemma.LetXbe a normed linear space,Ybe a closed proper subspace ofXand α be a real number with 0 < α < 1. Then there exists anxinXwith |x| = 1 such that |x−y| > α for allyinY.^{[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.

## 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 *x*_{1} 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.

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 *c*_{1}, ..., *c _{n}* ∈

*C*such that

We claim that the finite dimensional subspace *Y* spanned by {*c _{i}*}, or equivalently, its closure, is

*X*. Since scalar multiplication is continuous, its enough to show

*C*⊂

*Y*. Now, by induction,

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 for 0 <

*k*< 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, the X is finite dimensional ( proof by contradiction).

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}