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
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
Metric reformulation
As usual, let
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
from the origin , 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
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
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 this desired distance away from the subspace.

The proof can be found in functional analysis texts such as Kreyszig.
Minimum distances
not satisfying the hypotheses
When
is trivial then it has no 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 consequence of
being impossible is that the inequality
holds if and only if equality
holds.
Reflexivity
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
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
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
is reflexive if and only if for every closed proper vector subspace
there is some vector
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
that is always at least a distance of
away from the subspace.
In a non-reflexive Banach space, such as the
Lebesgue space of all bounded sequences, Riesz’s lemma does not hold for
.
Since every finite dimensional normed space is a reflexive Banach space, 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. The "perpendicular" vector may be found pictorially by drawing a unit sphere that is
supported by
at the origin.
For example, if the reflexive Banach space
is endowed with the usual
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
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
).
Some consequences
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 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
).
This sequence can be constructed by
induction for any constant
Start by picking any element
from the unit sphere. Let
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
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
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
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
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
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 ...
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
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
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
is compact if and only if
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