In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Banach–Stone theorem is a classical result in the theory of
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s on
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s, named after 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, mathematical structure, structure, space, Mathematica ...
s
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
and
Marshall Stone.
In brief, the Banach–Stone theorem allows one to recover a
compact Hausdorff space ''X'' from the Banach space structure of the space ''C''(''X'') of continuous real- or complex-valued functions on ''X''. If one is allowed to invoke the algebra structure of ''C''(''X'') this is easy – we can identify ''X'' with the
spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
of ''C''(''X''), the set of algebra homomorphisms into the scalar field, equipped with the weak*-topology inherited from the dual space ''C''(''X'')*. The Banach-Stone theorem avoids reference to multiplicative structure by recovering ''X'' from the extreme points of the unit ball of ''C''(''X'')*.
Statement
For a
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 ...
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
''X'', let ''C''(''X'') denote the
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 ...
of continuous real- or complex-valued
functions on ''X'', equipped with the
supremum norm
In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when t ...
‖·‖
∞.
Given compact Hausdorff spaces ''X'' and ''Y'', suppose ''T'' : ''C''(''X'') → ''C''(''Y'') is a
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
linear isometry. Then there exists a
homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
''φ'' : ''Y'' → ''X'' and a function ''g'' ∈ ''C''(''Y'') with
:
such that
:
The case where ''X'' and ''Y'' are compact
metric spaces is due to Banach, while the extension to compact Hausdorff spaces is due to Stone.
[Theorem 83 of ] In fact, they both prove a slight generalization—they do not assume that ''T'' is linear, only that it is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
in the sense of metric spaces, and use the
Mazur–Ulam theorem to show that ''T'' is affine, and so
is a linear isometry.
Generalizations
The Banach–Stone theorem has some generalizations for vector-valued continuous functions on compact, Hausdorff topological spaces. For example, if ''E'' 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 ...
with trivial
centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...
and ''X'' and ''Y'' are compact, then every linear isometry of ''C''(''X''; ''E'') onto ''C''(''Y''; ''E'') is a
strong Banach–Stone map.
A similar technique has also been used to recover a space ''X'' from the extreme points of the duals of some other spaces of functions on ''X''.
The noncommutative analog of the Banach-Stone theorem is the folklore theorem that two unital C*-algebras are isomorphic if and only if they are completely isometric (i.e., isometric at all matrix levels). Mere isometry is not enough, as shown by the existence of a C*-algebra that is not isomorphic to its opposite algebra (which trivially has the same Banach space structure).
See also
*
References
*
*
{{DEFAULTSORT:Banach-Stone theorem
Theory of continuous functions
Operator theory
Theorems in functional analysis