Continuous functions on a compact Hausdorff space

<p>In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, and especially <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a fundamental role is played by the space of <a href="/facts/Continuous_functions/sKbl02pB">continuous functions</a> on a <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 with values in the <a href="/facts/Real_numbers/R02gw5Pb">real</a> or <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a>.  This space, denoted by 
  
    
      
        
          
            C
          
        
        (
        X
        )
        ,
      
    
    {\displaystyle {\mathcal {C}}(X),}
  
 is a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> with respect to the pointwise addition of functions and scalar multiplication by constants.  It is, moreover, a <a href="/facts/Normed_space/rmDljfyE">normed space</a> with norm defined by

‖
        f
        ‖
        =
        
          sup
          
            x
            ∈
            X
          
        
        
          |
        
        f
        (
        x
        )
        
          |
        
        ,
      
    
    {\displaystyle \|f\|=\sup _{x\in X}|f(x)|,}

the <a href="/facts/Uniform_norm/5yIkjNE3">uniform norm</a>.  The uniform norm defines the <a href="/facts/Topology/2EqW47cU">topology</a> of <a href="/facts/Uniform_convergence/ecz9eNWn">uniform convergence</a> of functions on 
  
    
      
        X
        .
      
    
    {\displaystyle X.}
  
  The space 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 is a <a href="/facts/Banach_algebra/3BTHdpg2">Banach algebra</a> with respect to this norm.(Rudin 1973, §11.3)
</p>

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

Continuous functions on a compact Hausdorff space