The space is infinite-dimensional whenever is an infinite space (since it separates points). Hence, in particular, it is generally not
locally compact.
The
Stone–Weierstrass theorem holds for In the case of real functions, if is a
subring of that contains all constants and separates points, then the
closure of is In the case of complex functions, the statement holds with the additional hypothesis that is closed under
complex conjugation.
If and are two compact Hausdorff spaces, and is a
homomorphism of algebras which commutes with complex conjugation, then is continuous. Furthermore, has the form for some continuous function In particular, if and are isomorphic as algebras, then and are
homeomorphic topological spaces.
Let be the space of
maximal ideals in Then there is a one-to-one correspondence between Δ and the points of Furthermore, can be identified with the collection of all complex homomorphisms Equip with the
initial topology with respect to this pairing with (that is, the
Gelfand transform). Then is homeomorphic to Δ equipped with this topology. (
Rudin 1973, §11.13)
A sequence in is
weaklyCauchy if and only if it is (uniformly) bounded in and pointwise convergent. In particular, is only weakly complete for a finite set.
The
Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of for some
Generalizations
The space of real or complex-valued continuous functions can be defined on any topological space In the non-compact case, however, is not in general a Banach space with respect to the uniform norm since it may contain unbounded functions. Hence it is more typical to consider the space, denoted here of bounded continuous functions on This is a Banach space (in fact a commutative Banach algebra with identity) with respect to the uniform norm. (
Hewitt & Stromberg 1965, Theorem 7.9)
It is sometimes desirable, particularly in
measure theory, to further refine this general definition by considering the special case when is a
locally compact Hausdorff space. In this case, it is possible to identify a pair of distinguished subsets of : (
Hewitt & Stromberg 1965, §II.7)
the subset of consisting of functions with
compact support. This is called the space of functions vanishing in a neighborhood of infinity.
the subset of consisting of functions such that for every there is a compact set such that for all This is called the space of functions vanishing at infinity.
The closure of is precisely In particular, the latter is a Banach space.
References
Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
Hewitt, Edwin; Stromberg, Karl (1965), Real and abstract analysis, Springer-Verlag.
The space is infinite-dimensional whenever is an infinite space (since it separates points). Hence, in particular, it is generally not
locally compact.
The
Stone–Weierstrass theorem holds for In the case of real functions, if is a
subring of that contains all constants and separates points, then the
closure of is In the case of complex functions, the statement holds with the additional hypothesis that is closed under
complex conjugation.
If and are two compact Hausdorff spaces, and is a
homomorphism of algebras which commutes with complex conjugation, then is continuous. Furthermore, has the form for some continuous function In particular, if and are isomorphic as algebras, then and are
homeomorphic topological spaces.
Let be the space of
maximal ideals in Then there is a one-to-one correspondence between Δ and the points of Furthermore, can be identified with the collection of all complex homomorphisms Equip with the
initial topology with respect to this pairing with (that is, the
Gelfand transform). Then is homeomorphic to Δ equipped with this topology. (
Rudin 1973, §11.13)
A sequence in is
weaklyCauchy if and only if it is (uniformly) bounded in and pointwise convergent. In particular, is only weakly complete for a finite set.
The
Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of for some
Generalizations
The space of real or complex-valued continuous functions can be defined on any topological space In the non-compact case, however, is not in general a Banach space with respect to the uniform norm since it may contain unbounded functions. Hence it is more typical to consider the space, denoted here of bounded continuous functions on This is a Banach space (in fact a commutative Banach algebra with identity) with respect to the uniform norm. (
Hewitt & Stromberg 1965, Theorem 7.9)
It is sometimes desirable, particularly in
measure theory, to further refine this general definition by considering the special case when is a
locally compact Hausdorff space. In this case, it is possible to identify a pair of distinguished subsets of : (
Hewitt & Stromberg 1965, §II.7)
the subset of consisting of functions with
compact support. This is called the space of functions vanishing in a neighborhood of infinity.
the subset of consisting of functions such that for every there is a compact set such that for all This is called the space of functions vanishing at infinity.
The closure of is precisely In particular, the latter is a Banach space.
References
Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
Hewitt, Edwin; Stromberg, Karl (1965), Real and abstract analysis, Springer-Verlag.