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. 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. 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). In modern language, this is the commutative case of the spectrum of a C*-algebra, and the Banach–Stone theorem can be seen as a functional analysis analog of the connection between a ring R and the spectrum of a ring Spec(R) in algebraic geometry. 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 ‖·‖∞. This is an algebra, called the algebra of scalars, under pointwise multiplication of functions. For a compact space X, Cb(X; R) is the same as C(X; R), the space of all continuous functions f : X → R. The algebra of scalars is a functional analysis analog of the ring of regular functions in algebraic geometry, there denoted O X {displaystyle {mathcal {O}}_{X}} . 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