Banach–Stone theorem

Banach–Stone theorem

In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.

tatement of the theorem

For a topological space "X", let "C"b("X"; R) denote the normed vector space of continuous, real-valued, bounded functions "f" : "X" → R equipped with the supremum norm ||·||∞. For a compact space "X", "C"b("X"; R) is the same as "C"("X"; R), the space of all continuous functions "f" : "X" → R.

Let "X" and "Y" be compact, Hausdorff spaces and let "T" : "C"("X"; R) → "C"("Y"; R) be a surjective linear isometry. Then there exists a homeomorphism "φ" : "Y" → "X" and "g" ∈ "C"("Y"; R) with

:| g(y) | = 1 mbox{ for all } y in Y

and

:(T f) (y) = g(y) f(varphi(y)) mbox{ for all } y in Y, f in C(X; mathbf{R}).

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 with trivial centralizer and "X" and "Y" are compact, then every linear isometry of "C"("X"; "E") onto "C"("Y"; "E") is a strong Banach–Stone map.

References

* cite journal
last = Araujo
first = Jesús
title = The noncompact Banach–Stone theorem
journal = J. Operator Theory
volume = 55
year = 2006
issue = 2
pages = 285–294
issn = 0379-4024 MathSciNet|id=2242851


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