In the
mathematical
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 ...
field of
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, norm, topology, etc.) and the linear functions defi ...
, the Eberlein–Šmulian theorem (named after
William Frederick Eberlein and
Witold Lwowitsch Schmulian) is a result that relates three different kinds of
weak
Weak may refer to:
Songs
* "Weak" (AJR song), 2016
* "Weak" (Melanie C song), 2011
* "Weak" (SWV song), 1993
* "Weak" (Skunk Anansie song), 1995
* "Weak", a song by Seether from '' Seether: 2002-2013''
Television episodes
* "Weak" (''Fear t ...
compactness
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
in 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 ...
.
Statement
Eberlein–Šmulian theorem: If ''X'' 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 ...
and ''A'' is a subset of ''X'', then the following statements are equivalent:
# each sequence of elements of ''A'' has a subsequence that is weakly convergent in ''X''
# each sequence of elements of ''A'' has a weak
cluster point in ''X''
# the weak closure of ''A'' is weakly compact.
A set ''A'' can be weakly compact in three different ways:
*
Sequential compactness: Every sequence from ''A'' has a convergent subsequence whose limit is in ''A''.
*
Limit point compactness: Every infinite subset of ''A'' has a
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
in ''A''.
*
Compactness
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
(or
Heine
Heine is both a surname and a given name of German origin. People with that name include:
People with the surname
* Albert Heine (1867–1949), German actor
* Alice Heine (1858–1925), American-born princess of Monaco
* Armand Heine (1818–188 ...
-
Borel compactness): Every open cover of ''A'' admits a finite subcover.
The Eberlein–Šmulian theorem states that the three are equivalent on a weak topology of a Banach space.
While this equivalence is true in general for a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, the weak topology is not metrizable in infinite dimensional vector spaces, and so the Eberlein–Šmulian theorem is needed.
Applications
The Eberlein–Šmulian theorem is important in the theory of
PDEs, and particularly in
Sobolev spaces. Many Sobolev spaces are
reflexive Banach spaces and therefore bounded subsets are weakly precompact by
Alaoglu's theorem. Thus the theorem implies that bounded subsets are weakly sequentially precompact, and therefore from every bounded sequence of elements of that space it is possible to extract a subsequence which is weakly converging in the space. Since many PDEs only have solutions in the weak sense, this theorem is an important step in deciding which spaces of weak solutions to use in solving a PDE.
See also
*
Banach–Alaoglu theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology.
A common p ...
*
Bishop–Phelps theorem
*
Mazur's lemma
*
James' theorem In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space X is reflexive if and only if every continuous linear functional on X attains its supremum on the closed unit ball in X.
A st ...
*
Goldstine theorem
References
Bibliography
*
* .
* .
* .
Banach spaces
Compactness theorems
Theorems in functional analysis
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