In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, uniformly convex spaces (or uniformly rotund spaces) are common examples of
reflexive Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
s. The concept of uniform convexity was first introduced by
James A. Clarkson
James Andrew Clarkson (7 February 1906 – 6 June 1970) was an American mathematician and professor of mathematics who specialized in number theory. He is known for proving inequalities in Hölder condition#Hölder spaces, Hölder spaces, and deri ...
in 1936.
Definition
A uniformly convex space is a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
such that, for every
there is some
such that for any two vectors with
and
the condition
:
implies that:
:
Intuitively, the center of a line segment inside the
unit ball
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) in a theatrical presentation
Music
* ''Unit'' (alb ...
must lie deep inside the unit ball unless the segment is short.
Properties
* The
unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...
can be replaced with the closed unit
ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
in the definition. Namely, a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
is uniformly convex
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
for every
there is some
so that, for any two vectors
and
in the closed unit ball (i.e.
and
) with
, one has
(note that, given
, the corresponding value of
could be smaller than the one provided by the original weaker definition).
The "if" part is trivial. Conversely, assume now that
is uniformly convex and that
are as in the statement, for some fixed
. Let
be the value of
corresponding to
in the definition of uniform convexity. We will show that
, with
.
If
then
and the claim is proved. A similar argument applies for the case
, so we can assume that
. In this case, since
, both vectors are nonzero, so we can let
and
. We have
and similarly
, so
and
belong to the unit sphere and have distance
. Hence, by our choice of
, we have
. It follows that
and the claim is proved.
* The
Milman–Pettis theorem
In mathematics, the Milman–Pettis theorem states that every uniformly convex space, uniformly convex Banach space is reflexive space, reflexive.
The theorem was proved independently by David Milman, D. Milman (1938) and B. J. Pettis (1939). Shiz ...
states that every uniformly convex
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
is
reflexive, while the converse is not true.
* Every uniformly convex
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
is a Radon-Riesz space, that is, if
is a sequence in a uniformly convex Banach space which converges weakly to
and satisfies
then
converges strongly to
, that is,
.
* A
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
is uniformly convex if and only if its dual
is
uniformly smooth.
* Every uniformly convex space is
strictly convex. Intuitively, the strict convexity means a stronger
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
whenever
are linearly independent, while the uniform convexity requires this inequality to be true uniformly.
Examples
* Every Hilbert space is uniformly convex.
* Every closed subspace of a uniformly convex Banach space is uniformly convex.
*
Hanner's inequalities
In mathematics, Hanner's inequalities are results in the theory of ''L'p'' spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of ''L'p'' spaces for ''p'' ∈ (1,& ...
imply that
L''p'' spaces