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 ...

— specifically, 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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

— an Asplund space or strong differentiability space is a type of

well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. Th ...

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 ...

. Asplund spaces were introduced in 1968 by the

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Edgar Asplund Edgar is a commonly used English given name, from an Anglo-Saxon name ''Eadgar'' (composed of '' ead'' "rich, prosperous" and '' gar'' "spear"). Like most Anglo-Saxon names, it fell out of use by the later medieval period; it was, however, re ...

, who was interested in the Fréchet differentiability properties of

Lipschitz function In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exi ...

s on Banach spaces.

Equivalent definitions

There are many equivalent definitions of what it means for a Banach space ''X'' to be an Asplund space: * ''X'' is Asplund if, and only if, every separable subspace ''Y'' of ''X'' has separable

continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...

''Y''^∗. * ''X'' is Asplund if, and only if, every

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...

on any

open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...

convex subset In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

''U'' of ''X'' is Fréchet differentiable at the points of a

dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

''G''_''δ''-subset of ''U''. * ''X'' is Asplund if, and only if, its dual space ''X''^∗ has the Radon–Nikodým property. This property was established by Namioka & Phelps in 1975 and Stegall in 1978. * ''X'' is Asplund if, and only if, every non-empty bounded subset of its dual space ''X''^∗ has weak-∗-slices of arbitrarily small diameter. * ''X'' is Asplund if and only if every non-empty weakly-∗

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

subset of the dual space ''X''^∗ is the weakly-∗ closed

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

of its weakly-∗ strongly

exposed point In mathematics, an exposed point of a convex set C is a point x\in C at which some continuous linear functional attains its strict maximum over C. Such a functional is then said to ''expose'' x. There can be many exposing functionals for x. The se ...

s. In 1975, Huff & Morris showed that this property is equivalent to the statement that every bounded, closed and convex subset of the dual space ''X''^∗ is closed convex hull of its extreme points.

Properties of Asplund spaces

* The class of Asplund spaces is closed under topological isomorphisms: that is, if ''X'' and ''Y'' are Banach spaces, ''X'' is Asplund, and ''X'' is

homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

to ''Y'', then ''Y'' is also an Asplund space. * Every closed

linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...

of an Asplund space is an Asplund space. * Every quotient space of an Asplund space is an Asplund space. * The class of Asplund spaces is closed under extensions: if ''X'' is a Banach space and ''Y'' is an Asplund subspace of ''X'' for which the quotient space ''X'' ⁄ ''Y'' is Asplund, then ''X'' is Asplund. * Every locally Lipschitz function on an open subset of an Asplund space is Fréchet differentiable at the points of some dense subset of its domain. This result was established by

Preiss Preiss is a Germanic surname, and may refer to: * Ferdinand Preiss (1882–1943), German sculptor * Balthazar Preiss (1765-1850), Austrian naturalist * Ludwig Preiss (1811–1883), German naturalist * Wolfgang Preiss (1910–2002), German actor * ...

in 1990 and has applications in optimization theory. * The following theorem from Asplund's original 1968 paper is a good example of why non-Asplund spaces are badly behaved: if ''X'' is not an Asplund space, then there exists an equivalent norm on ''X'' that fails to be Fréchet differentiable at every point of ''X''. * In 1976, Ekeland & Lebourg showed that if ''X'' is a Banach space that has an equivalent norm that is Fréchet differentiable away from the origin, then ''X'' is an Asplund space. However, in 1990, Haydon gave an example of an Asplund space that does not have an equivalent norm that is Gateaux differentiable away from the origin.

References

* * * * * * * {{Topological vector spaces Banach spaces Functional analysis