Riesz's lemma (after

_{∞} 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. A

online proof from Prof. Paul Garrett

is available.

_{1} from the unit sphere. Pick ''x_{n}'' from the unit sphere such that
:$d(x\_n,\; Y\_)\; >\; \backslash alpha$ for a constant 0 < ''α'' < 1, where ''Y''_{''n''−1} is the linear span of and $d(x\_n,\; Y)\; =\; \backslash inf\_\; ,\; x\_n\; -\; y,$.
Clearly contains no convergent subsequence and the noncompactness of the unit ball follows.
More generally, if a _{1}, ..., ''c_{n}'' ∈ ''C'' such that
:$C\; \backslash sub\; \backslash bigcup\_^n\; \backslash ;\; \backslash left(\; c\_i\; +\; \backslash frac\; C\; \backslash right).$
We claim that the finite dimensional subspace ''Y'' spanned by is dense in ''X'', or equivalently, its closure is ''X''. Since ''X'' is the union of scalar multiples of ''C'', it is sufficient to show that ''C'' ⊂ ''Y''. Now, by induction,
:$C\; \backslash sub\; Y\; +\; \backslash frac\; C$
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.https://www.emis.de/journals/PM/51f2/pm51f205.pdf

Frigyes Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathematic ...

) is a lemma
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
* Headword, under which a set of related dict ...

in functional analysis
200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.
Functional analysis is a branch of mathemat ...

. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

is dense
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...

. 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

Riesz's Lemma. Let ''X'' be a normed space, ''Y'' be a closed proper subspace of ''X'' and α be a real number with 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

online proof from Prof. Paul Garrett

is available.

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 with $,\; x\_n\; -\; x\_m,\; >\; \backslash alpha$ for 0 < ''α'' < 1. This is useful in showing the non-existence of certain measures on infinite-dimensionalBanach space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s. Riesz's lemma also shows that the identity operator on a Banach space ''X'' is compact if and only if ''X'' is finite-dimensional.
One can also use this lemma to characterize finite dimensional normed spaces: if X is a normed vector space, then X is finite dimensional if and only if the closed unit ball in X is compact.
Characterization of finite dimension

Riesz's lemma can be applied directly to show that theunit ball
Unit may refer to:
Arts and entertainment
* UNIT
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) i ...

of an infinite-dimensional normed space ''X'' is never compact
Compact as used in politics may refer broadly to a pact
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations
International relations (IR), international affairs (IA) or internationa ...

: Take an element ''x''topological vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

''X'' is locally compact In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

, 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 ''c''See also

* F. Riesz's theoremReferences

* {{Banach spaces Functional analysis Lemmas in analysis