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In mathematics, Riesz's lemma (after Frigyes Riesz) is a lemma in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. It specifies (often easy to check) conditions that guarantee that a subspace in a
normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
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 In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
.


Statement

If X is a reflexive
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
then this conclusion is also true when \alpha = 1. Metric reformulation As usual, let d(x, y) := \, x - y\, denote the canonical
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
induced by the norm, call the set \ of all vectors that are a distance of 1 from the origin , and denote the distance from a point u to the set Y \subseteq X by d(u, Y) ~:=~ \inf_ d(u, y) ~=~ \inf_ \, u - y\, . The inequality \alpha \leq d(u, Y) holds if and only if \, u - y\, \geq \alpha for all y \in Y, and it formally expresses the notion that the distance between u and Y is at least \alpha. Because every vector subspace (such as Y) contains the origin 0, substituting y := 0 in this
infimum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
shows that d(u, Y) \leq \, u\, for every vector u \in X. In particular, d(u, Y) \leq 1 when \, u\, = 1 is a unit vector. Using this new notation, the conclusion of Riesz's lemma may be restated more succinctly as: \alpha \leq d(u, Y) \leq 1 = \, u\, holds for some u \in X. 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 X, for any desired minimum distance \alpha less than 1, there exists some vector in the unit sphere of X that is this desired distance away from the subspace. The proof can be found in functional analysis texts such as Kreyszig. Minimum distances \alpha not satisfying the hypotheses When X = \ is trivial then it has no vector subspace Y, and so Riesz's lemma holds vacuously for all real numbers \alpha \in \R. The remainder of this section will assume that X \neq \, which guarantees that a unit vector exists. The inclusion of the hypotheses 0 < \alpha < 1 can be explained by considering the three cases: \alpha \leq 0, \alpha = 1, and \alpha > 1. The lemma holds when \alpha \leq 0 since every unit vector u \in X satisfies the conclusion \alpha \leq 0 \leq d(u, Y) \leq 1 = \, u\, . The hypotheses 0 < \alpha is included solely to exclude this trivial case and is sometimes omitted from the lemma's statement. Riesz's lemma is always false when \alpha > 1 because for every unit vector u \in X, the required inequality \, u - y\, \geq \alpha fails to hold for y := 0 \in Y (since \, u - 0\, = 1 < \alpha). Another consequence of d(u, Y) > 1 being impossible is that the inequality d(u, Y) \geq 1 holds if and only if equality d(u, Y) = 1 holds.


Reflexivity

This leaves only the case \alpha = 1 for consideration, in which case the statement of Riesz’s lemma becomes: :For every closed proper vector subspace Y of X, there exists some vector u of unit norm that satisfies d(u, Y) = 1. When X is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
, then this statement is true if and only if X is a
reflexive space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomo ...
. Explicitly, a Banach space X is reflexive if and only if for every closed proper vector subspace Y, there is some vector u on the
unit sphere In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...
of X that is always at least a distance of 1 = d(u, Y) away from the subspace. In a non-reflexive Banach space, such as the Lebesgue space \ell_\infty(\N) of all bounded sequences, Riesz’s lemma does not hold for \alpha = 1. Since every finite dimensional normed space is a reflexive Banach space, Riesz’s lemma does holds for \alpha = 1 when the normed space is finite-dimensional, as will now be shown. When the dimension of X is finite then the closed unit ball B \subseteq X is compact. Since the distance function d(\cdot, Y) is continuous, its image on the closed unit ball B must be a compact subset of the real line, proving the claim. The "perpendicular" vector may be found pictorially by drawing a unit sphere that is supported by Y at the origin. For example, if the reflexive Banach space X = \Reals^3 is endowed with the usual \, \cdot\, _2
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...
and if Y = \Reals \times \Reals \times \ is the x\texty plane then the points u = (0, 0, \pm 1) satisfy the conclusion d(u, Y) = 1. If Z = \ \times \Reals is z-axis then every point u belonging to the unit circle in the x\texty plane satisfies the conclusion d(u, Z) = 1. But if X = \Reals^3 was endowed with the \, \cdot\, _1 taxicab norm (instead of the Euclidean norm), then the conclusion d(u, Z) = 1 would be satisfied by every point u = (x, y, 0) belonging to the “diamond” , x, + , y, = 1 in the x\texty plane (a square with vertices at (\pm 1, 0, 0) and (0, \pm 1, 0)).


Some consequences

Riesz's lemma guarantees that for any given 0 < \alpha < 1, every infinite-dimensional normed space contains a sequence x_1, x_2, \ldots of (distinct) unit vectors satisfying \, x_n - x_m\, > \alpha for m \neq n; or stated in plain English, these vectors are all separated from each other by a distance of more than \alpha 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 mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
in \Reals^2). This sequence can be constructed by induction for any constant 0 < \alpha < 1. Start by picking any element x_1 from the unit sphere. Let Y_ be the
linear span In mathematics, the linear span (also called the linear hull or just span) of a set S of elements of a vector space V is the smallest linear subspace of V that contains S. It is the set of all finite linear combinations of the elements of , and ...
of \ and (using Riesz's lemma) pick x_n from the unit sphere such that :d\left(x_n, Y_\right) > \alpha where d(x_n, Y) = \inf_ \, x_n - y\, . This sequence x_1, x_2, \ldots contains no convergent subsequence, which implies that the closed unit ball is not compact.


Characterization of finite dimension

Riesz's lemma can be applied directly to show that the
unit ball Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
of an infinite-dimensional normed space X is never
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
. This can be used 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. More generally, if a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
X is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
, 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 the origin in X. By compactness, there are c_1, \ldots, c_n \in C such that C ~\subseteq~ \left(c_1 + \tfrac C\right) \cup \cdots \cup \left(c_n + \tfrac C\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 \subseteq Y. By induction, for every m, C ~\subseteq~ Y + \frac C. 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 In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
see .


Spectral theory

The spectral properties of compact operators acting on a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
are similar to those of matrices. Riesz's lemma is essential in establishing this fact.


Other applications

As detailed in the article on
infinite-dimensional Lebesgue measure In mathematics, an infinite-dimensional Lebesgue measure is a measure defined on infinite-dimensional normed vector spaces, such as Banach spaces, which resembles the Lebesgue measure used in finite-dimensional spaces. However, the traditiona ...
, this is useful in showing the non-existence of certain measures on infinite-dimensional
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
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.


See also

* *
James's Theorem In mathematics, particularly functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product s ...
—a characterization of reflexivity given by a condition on the unit ball


References

* * * * *


Further reading

* https://mathoverflow.net/questions/470438/a-variation-of-the-riesz-lemma {{Banach spaces Functional analysis Lemmas in mathematical analysis Normed spaces