Continuous functions on a compact Hausdorff space

<h2 id="properties">Properties</h2>
<ul><li>By <a href="/facts/Urysohn%27s_lemma/FmC5kMdG">Urysohn's lemma</a>, 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 <a href="/facts/Separating_set/I8VjYAgQ">separates points</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
: If 
  
    
      
        x
        ,
        y
        ∈
        X
      
    
    {\displaystyle x,y\in X}
  
 are distinct points, then there is an 
  
    
      
        f
        ∈
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle f\in {\mathcal {C}}(X)}
  
 such that 
  
    
      
        f
        (
        x
        )
        ≠
        f
        (
        y
        )
        .
      
    
    {\displaystyle f(x)\neq f(y).}
  
</li>
<li>The space 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 is infinite-dimensional whenever 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is an infinite space (since it separates points).  Hence, in particular, it is generally not <a href="/facts/Locally_compact/hesalT7q">locally compact</a>.</li>
<li>The <a href="/facts/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem/vtJ7grhH">Riesz–Markov–Kakutani representation theorem</a> gives a characterization of the <a href="/facts/Continuous_dual_space/4Uw4Knz1">continuous dual space</a> of 
  
    
      
        
          
            C
          
        
        (
        X
        )
        .
      
    
    {\displaystyle {\mathcal {C}}(X).}
  
  Specifically, this dual space is the space of <a href="/facts/Radon_measure/21adq3XN">Radon measures</a> on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 (regular <a href="/facts/Borel_measure/8cYR8Ysw">Borel measures</a>), denoted by 
  
    
      
        rca
        ⁡
        (
        X
        )
        .
      
    
    {\displaystyle \operatorname {rca} (X).}
  
  This space, with the norm given by the <a href="/facts/Total_variation/tyY5fTz9">total variation</a> of a measure, is also a Banach space belonging to the class of <a href="/facts/Ba_space/CUXe79cw">ba spaces</a>. (Dunford & Schwartz 1958, §IV.6.3)</li>
<li><a href="/facts/Positive_linear_functional/gEGtP6AQ">Positive linear functionals</a> on 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 correspond to (positive) <a href="/facts/Regular_measure/zqVDoMv4">regular</a> <a href="/facts/Borel_measure/8cYR8Ysw">Borel measures</a> on 
  
    
      
        X
        ,
      
    
    {\displaystyle X,}
  
 by a different form of the Riesz representation theorem. (Rudin 1966, Chapter 2)</li>
<li>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is infinite, then 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 is not <a href="/facts/Reflexive_space/g7Cjbw0S">reflexive</a>, nor is it <a href="/facts/Weak_topology/XJASE9Up">weakly</a> <a href="/facts/Complete_space/lsckmU8G">complete</a>.</li>
<li>The <a href="/facts/Arzel%C3%A0%E2%80%93Ascoli_theorem/eqSFHXbo">Arzelà–Ascoli theorem</a> holds: A subset 
  
    
      
        K
      
    
    {\displaystyle K}
  
 of 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 is <a href="/facts/Relatively_compact/z5uIqMDp">relatively compact</a> if and only if it is <a href="/facts/Bounded_set/yCldjtoy">bounded</a> in the norm of 
  
    
      
        
          
            C
          
        
        (
        X
        )
        ,
      
    
    {\displaystyle {\mathcal {C}}(X),}
  
 and <a href="/facts/Equicontinuous/cKjFQWD6">equicontinuous</a>.</li>
<li>The <a href="/facts/Stone%E2%80%93Weierstrass_theorem/QB07Nwba">Stone–Weierstrass theorem</a> holds for 
  
    
      
        
          
            C
          
        
        (
        X
        )
        .
      
    
    {\displaystyle {\mathcal {C}}(X).}
  
  In the case of real functions, if 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is a <a href="/facts/Subring/M0perybt">subring</a> of 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 that contains all constants and separates points, then the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is 
  
    
      
        
          
            C
          
        
        (
        X
        )
        .
      
    
    {\displaystyle {\mathcal {C}}(X).}
  
  In the case of complex functions, the statement holds with the additional hypothesis that 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is closed under <a href="/facts/Complex_conjugation/7TEaoQDe">complex conjugation</a>.</li>
<li>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 are two compact Hausdorff spaces, and 
  
    
      
        F
        :
        
          
            C
          
        
        (
        X
        )
        →
        
          
            C
          
        
        (
        Y
        )
      
    
    {\displaystyle F:{\mathcal {C}}(X)\to {\mathcal {C}}(Y)}
  
 is a <a href="/facts/Homomorphism/0gyqdEse">homomorphism</a> of algebras which commutes with complex conjugation, then 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is continuous.  Furthermore, 
  
    
      
        F
      
    
    {\displaystyle F}
  
 has the form 
  
    
      
        F
        (
        h
        )
        (
        y
        )
        =
        h
        (
        f
        (
        y
        )
        )
      
    
    {\displaystyle F(h)(y)=h(f(y))}
  
 for some continuous function 
  
    
      
        f
        :
        Y
        →
        X
        .
      
    
    {\displaystyle f:Y\to X.}
  
  In particular, if 
  
    
      
        C
        (
        X
        )
      
    
    {\displaystyle C(X)}
  
 and 
  
    
      
        C
        (
        Y
        )
      
    
    {\displaystyle C(Y)}
  
 are isomorphic as algebras, then 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 are <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphic</a> topological spaces.</li>
<li>Let 
  
    
      
        Δ
      
    
    {\displaystyle \Delta }
  
 be the space of <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideals</a> in 
  
    
      
        
          
            C
          
        
        (
        X
        )
        .
      
    
    {\displaystyle {\mathcal {C}}(X).}
  
  Then there is a one-to-one correspondence between Δ and the points of 
  
    
      
        X
        .
      
    
    {\displaystyle X.}
  
  Furthermore, 
  
    
      
        Δ
      
    
    {\displaystyle \Delta }
  
 can be identified with the collection of all complex homomorphisms 
  
    
      
        
          
            C
          
        
        (
        X
        )
        →
        
          C
        
        .
      
    
    {\displaystyle {\mathcal {C}}(X)\to \mathbb {C} .}
  
  Equip 
  
    
      
        Δ
      
    
    {\displaystyle \Delta }
  
with the <a href="/facts/Initial_topology/bNRdJM81">initial topology</a> with respect to this pairing with 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 (that is, the <a href="/facts/Gelfand_transform/QV54f7Jm">Gelfand transform</a>).  Then 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is homeomorphic to Δ equipped with this topology. (Rudin 1973, §11.13)</li>
<li>A sequence in 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 is <a href="/facts/Weak_topology/XJASE9Up">weakly</a> <a href="/facts/Cauchy_sequence/7Q5c8TuF">Cauchy</a> if and only if it is (uniformly) bounded in 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 and pointwise convergent.  In particular, 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 is only weakly complete for 
  
    
      
        X
      
    
    {\displaystyle X}
  
 a finite set.</li>
<li>The <a href="/facts/Vague_topology/jcHas6W8">vague topology</a> is the <a href="/facts/Weak*_topology/XJASE9Up">weak* topology</a> on the dual of 
  
    
      
        
          
            C
          
        
        (
        X
        )
        .
      
    
    {\displaystyle {\mathcal {C}}(X).}
  
</li>
<li>The <a href="/facts/Banach%E2%80%93Alaoglu_theorem/NILk5nDm">Banach–Alaoglu theorem</a> implies that any normed space is isometrically isomorphic to a subspace of 
  
    
      
        C
        (
        X
        )
      
    
    {\displaystyle C(X)}
  
 for some 
  
    
      
        X
        .
      
    
    {\displaystyle X.}
  
</li></ul>
<h2 id="generalizations">Generalizations</h2>
<p>The space 
  
    
      
        C
        (
        X
        )
      
    
    {\displaystyle C(X)}
  
 of real or complex-valued continuous functions can be defined on any topological space 
  
    
      
        X
        .
      
    
    {\displaystyle X.}
  
  In the non-compact case, however, 
  
    
      
        C
        (
        X
        )
      
    
    {\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 
  
    
      
        
          C
          
            B
          
        
        (
        X
        )
      
    
    {\displaystyle C_{B}(X)}
  
 of bounded continuous functions on 
  
    
      
        X
        .
      
    
    {\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)
</p><p>It is sometimes desirable, particularly in <a href="/facts/Measure_theory/yCq7zlI4">measure theory</a>, to further refine this general definition by considering the special case when 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a <a href="/facts/Locally_compact/hesalT7q">locally compact</a> Hausdorff space.  In this case, it is possible to identify a pair of distinguished subsets of 
  
    
      
        
          C
          
            B
          
        
        (
        X
        )
      
    
    {\displaystyle C_{B}(X)}
  
: (Hewitt & Stromberg 1965, §II.7)
</p>
<ul><li>
  
    
      
        
          C
          
            00
          
        
        (
        X
        )
        ,
      
    
    {\displaystyle C_{00}(X),}
  
 the subset of 
  
    
      
        C
        (
        X
        )
      
    
    {\displaystyle C(X)}
  
 consisting of functions with <a href="/facts/Compact_support/BTuMGbsb">compact support</a>.  This is called the space of functions vanishing in a neighborhood of infinity.</li>
<li>
  
    
      
        
          C
          
            0
          
        
        (
        X
        )
        ,
      
    
    {\displaystyle C_{0}(X),}
  
 the subset of 
  
    
      
        C
        (
        X
        )
      
    
    {\displaystyle C(X)}
  
 consisting of functions such that for every 
  
    
      
        r
        >
        0
        ,
      
    
    {\displaystyle r>0,}
  
 there is a compact set 
  
    
      
        K
        ⊆
        X
      
    
    {\displaystyle K\subseteq X}
  
 such that 
  
    
      
        
          |
        
        f
        (
        x
        )
        
          |
        
        <
        r
      
    
    {\displaystyle |f(x)|<r}
  
 for all 
  
    
      
        x
        ∈
        X
        ∖
        K
        .
      
    
    {\displaystyle x\in X\backslash K.}
  
  This is called the space of functions <a href="/facts/Vanish_at_infinity/nCAl6pYS">vanishing at infinity</a>.</li></ul>
<p>The closure of 
  
    
      
        
          C
          
            00
          
        
        (
        X
        )
      
    
    {\displaystyle C_{00}(X)}
  
 is precisely 
  
    
      
        
          C
          
            0
          
        
        (
        X
        )
        .
      
    
    {\displaystyle C_{0}(X).}
  
  In particular, the latter is a Banach space.
</p>

<ul><li>Dunford, N.; Schwartz, J.T. (1958), <i>Linear operators, Part I</i>, Wiley-Interscience.</li>
<li>Hewitt, Edwin; Stromberg, Karl (1965), <i>Real and abstract analysis</i>, Springer-Verlag.</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1991). <a href="https://archive.org/details/functionalanalys00rudi"><i>Functional Analysis</i></a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/facts/McGraw-Hill_Science/Engineering/Math/srSyQSsY">McGraw-Hill Science/Engineering/Math</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054236-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21163277">21163277</a>.</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1966), <i>Real and complex analysis</i>, McGraw-Hill, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-054234-1.</li></ul>

Continuous functions on a compact Hausdorff space open-in-new

Continuous functions on a compact Hausdorff space