In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space  with values in the real or complex numbers.  This space, denoted by  is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants.  It is, moreover, a normed space with norm defined by

the uniform norm.  The uniform norm defines the topology of uniform convergence of functions on   The space  is a Banach algebra with respect to this norm.

Properties 

 By Urysohn's lemma,  separates points of : If  are distinct points, then there is an  such that 
 The space  is infinite-dimensional whenever  is an infinite space (since it separates points).  Hence, in particular, it is generally not locally compact.
 The Riesz–Markov–Kakutani representation theorem gives a characterization of the continuous dual space of   Specifically, this dual space is the space of Radon measures on  (regular Borel measures), denoted by   This space, with the norm given by the total variation of a measure, is also a Banach space belonging to the class of ba spaces. 
 Positive linear functionals on  correspond to (positive) regular Borel measures on  by a different form of the Riesz representation theorem. 
 If  is infinite, then  is not reflexive, nor is it weakly complete.
 The Arzelà–Ascoli theorem holds: A subset  of  is relatively compact if and only if it is bounded in the norm of  and equicontinuous.
 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. 
 A sequence in  is weakly Cauchy if and only if it is (uniformly) bounded in  and pointwise convergent.  In particular,  is only weakly complete for  a finite set.
 The vague topology is the weak* topology on the dual of 
 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. 

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 : 

  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 

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Banach spaces
Complex analysis
Theory of continuous functions
Functional analysis
Real analysis
Types of functions