• By Urysohn's lemma, ${\displaystyle {\mathcal {C}}(X)}$  separates points of ${\displaystyle X}$ : If ${\displaystyle x,y\in X}$  are distinct points, then there is an ${\displaystyle f\in {\mathcal {C}}(X)}$  such that ${\displaystyle f(x)\neq f(y).}$ 
• The space ${\displaystyle {\mathcal {C}}(X)}$  is infinite-dimensional whenever ${\displaystyle X}$  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 ${\displaystyle {\mathcal {C}}(X).}$  Specifically, this dual space is the space of Radon measures on ${\displaystyle X}$  (regular Borel measures), denoted by ${\displaystyle \operatorname {rca} (X).}$  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. (Dunford & Schwartz 1958, §IV.6.3)
• Positive linear functionals on ${\displaystyle {\mathcal {C}}(X)}$  correspond to (positive) regular Borel measures on ${\displaystyle X,}$  by a different form of the Riesz representation theorem. (Rudin 1966, Chapter 2)
• If ${\displaystyle X}$  is infinite, then ${\displaystyle {\mathcal {C}}(X)}$  is not reflexive, nor is it weakly complete.
• The Arzelà–Ascoli theorem holds: A subset ${\displaystyle K}$  of ${\displaystyle {\mathcal {C}}(X)}$  is relatively compact if and only if it is bounded in the norm of ${\displaystyle {\mathcal {C}}(X),}$  and equicontinuous.
• The Stone–Weierstrass theorem holds for ${\displaystyle {\mathcal {C}}(X).}$  In the case of real functions, if ${\displaystyle A}$  is a subring of ${\displaystyle {\mathcal {C}}(X)}$  that contains all constants and separates points, then the closure of ${\displaystyle A}$  is ${\displaystyle {\mathcal {C}}(X).}$  In the case of complex functions, the statement holds with the additional hypothesis that ${\displaystyle A}$  is closed under complex conjugation.
• If ${\displaystyle X}$  and ${\displaystyle Y}$  are two compact Hausdorff spaces, and ${\displaystyle F:{\mathcal {C}}(X)\to {\mathcal {C}}(Y)}$  is a homomorphism of algebras which commutes with complex conjugation, then ${\displaystyle F}$  is continuous. Furthermore, ${\displaystyle F}$  has the form ${\displaystyle F(h)(y)=h(f(y))}$  for some continuous function ${\displaystyle f:Y\to X.}$  In particular, if ${\displaystyle C(X)}$  and ${\displaystyle C(Y)}$  are isomorphic as algebras, then ${\displaystyle X}$  and ${\displaystyle Y}$  are homeomorphic topological spaces.
• Let ${\displaystyle \Delta }$  be the space of maximal ideals in ${\displaystyle {\mathcal {C}}(X).}$  Then there is a one-to-one correspondence between Δ and the points of ${\displaystyle X.}$  Furthermore, ${\displaystyle \Delta }$  can be identified with the collection of all complex homomorphisms ${\displaystyle {\mathcal {C}}(X)\to \mathbb {C} .}$  Equip ${\displaystyle \Delta }$ with the initial topology with respect to this pairing with ${\displaystyle {\mathcal {C}}(X)}$  (that is, the Gelfand transform). Then ${\displaystyle X}$  is homeomorphic to Δ equipped with this topology. (Rudin 1973, §11.13)
• A sequence in ${\displaystyle {\mathcal {C}}(X)}$  is weakly Cauchy if and only if it is (uniformly) bounded in ${\displaystyle {\mathcal {C}}(X)}$  and pointwise convergent. In particular, ${\displaystyle {\mathcal {C}}(X)}$  is only weakly complete for ${\displaystyle X}$  a finite set.
• The vague topology is the weak* topology on the dual of ${\displaystyle {\mathcal {C}}(X).}$ 
• The Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of ${\displaystyle C(X)}$  for some ${\displaystyle X.}$ 

The space ${\displaystyle C(X)}$  of real or complex-valued continuous functions can be defined on any topological space ${\displaystyle X.}$  In the non-compact case, however, ${\displaystyle C(X)}$  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 ${\displaystyle C_{B}(X)}$  of bounded continuous functions on ${\displaystyle X.}$  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 ${\displaystyle X}$  is a locally compact Hausdorff space. In this case, it is possible to identify a pair of distinguished subsets of ${\displaystyle C_{B}(X)}$ : (Hewitt & Stromberg 1965, §II.7)

• ${\displaystyle C_{00}(X),}$  the subset of ${\displaystyle C(X)}$  consisting of functions with compact support. This is called the space of functions vanishing in a neighborhood of infinity.
• ${\displaystyle C_{0}(X),}$  the subset of ${\displaystyle C(X)}$  consisting of functions such that for every ${\displaystyle r>0,}$  there is a compact set ${\displaystyle K\subseteq X}$  such that ${\displaystyle |f(x)|<r}$  for all ${\displaystyle x\in X\backslash K.}$  This is called the space of functions vanishing at infinity.

The closure of ${\displaystyle C_{00}(X)}$  is precisely ${\displaystyle C_{0}(X).}$  In particular, the latter is a Banach space.

• Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
• Hewitt, Edwin; Stromberg, Karl (1965), Real and abstract analysis, Springer-Verlag.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Rudin, Walter (1966), Real and complex analysis, McGraw-Hill, ISBN 0-07-054234-1.