Banach-Stone theorem

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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.

[edit] Statement of the theorem

For a topological space X, let Cb(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, Cb(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}).

[edit] 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.

[edit] References

  • Araujo, Jesús (2006). "The noncompact Banach-Stone theorem". J. Operator Theory 55 (2): 285–294. ISSN 0379-4024.  MR2242851