Banach-Stone theorem

They are used, for example, in generalizations of the Banach-Stone theorem.

More significantly, the Banach-Stone theorem suggests the philosophy that one can replace a space (a geometric notion) by an algebra, with no loss.

The Banach-Stone theorem has some generalizations for vector-valued continuous functions on compact, Hausdorff topological spaces.

In brief, the Banach-Stone theorem allows one to recover a compact Hausdorff space from the algebra of scalars (the bounded continuous functions on the space).

The whole topological information of a smooth manifold is encoded in the algebraic properties of its -algebra of smooth functions as in the 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.