Square-integrable function

<h2 id="properties">Properties</h2>
<p>The square integrable functions (in the sense mentioned in which a "function" actually means an <a href="/facts/Equivalence_class/1d2sh0TB">equivalence class</a> of functions that are equal almost everywhere) form an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> with <a href="/facts/Inner_product/HoyjElEy">inner product</a> given by

⟨
        f
        ,
        g
        ⟩
        =
        
          ∫
          
            A
          
        
        f
        (
        x
        )
        
          
            
              g
              (
              x
              )
            
            ¯
          
        
        
        
          d
        
        x
        ,
      
    
    {\displaystyle \langle f,g\rangle =\int _{A}f(x){\overline {g(x)}}\,\mathrm {d} x,}

where
</p>
<ul><li>
  
    
      
        f
      
    
    {\displaystyle f}
  
 and 
  
    
      
        g
      
    
    {\displaystyle g}
  
 are square integrable functions,</li>
<li>
  
    
      
        
          
            
              f
              (
              x
              )
            
            ¯
          
        
      
    
    {\displaystyle {\overline {f(x)}}}
  
 is the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of 
  
    
      
        f
        (
        x
        )
        ,
      
    
    {\displaystyle f(x),}
  
</li>
<li>
  
    
      
        A
      
    
    {\displaystyle A}
  
 is the set over which one integrates—in the first definition (given in the introduction above), 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is 
  
    
      
        (
        −
        ∞
        ,
        +
        ∞
        )
      
    
    {\displaystyle (-\infty ,+\infty )}
  
, in the second, 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
.</li></ul>
<p>Since 
  
    
      
        
          |
        
        a
        
          
            |
          
          
            2
          
        
        =
        a
        ⋅
        
          
            a
            ¯
          
        
      
    
    {\displaystyle |a|^{2}=a\cdot {\overline {a}}}
  
, square integrability is the same as saying

⟨
        f
        ,
        f
        ⟩
        <
        ∞
        .
        
      
    
    {\displaystyle \langle f,f\rangle <\infty .\,}

</p><p>It can be shown that square integrable functions form a <a href="/facts/Complete_metric_space/lsckmU8G">complete metric space</a> under the metric induced by the inner product defined above.
A complete metric space is also called a <a href="/facts/Cauchy_space/1Q8kEebw">Cauchy space</a>, because sequences in such metric spaces converge if and only if they are <a href="/facts/Cauchy_sequence/7Q5c8TuF">Cauchy</a>.
A space that is complete under the metric induced by a norm is a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>.
Therefore, the space of square integrable functions is a Banach space, under the metric induced by the norm, which in turn is induced by the inner product.
As we have the additional property of the inner product, this is specifically a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>, because the space is complete under the metric induced by the inner product.
</p><p>This inner product space is conventionally denoted by 
  
    
      
        
          (
          
            
              L
              
                2
              
            
            ,
            ⟨
            ⋅
            ,
            ⋅
            
              ⟩
              
                2
              
            
          
          )
        
      
    
    {\displaystyle \left(L_{2},\langle \cdot ,\cdot \rangle _{2}\right)}
  
 and many times abbreviated as 
  
    
      
        
          L
          
            2
          
        
        .
      
    
    {\displaystyle L_{2}.}

Note that 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L_{2}}
  
 denotes the set of square integrable functions, but no selection of metric, norm or inner product are specified by this notation.
The set, together with the specific inner product 
  
    
      
        ⟨
        ⋅
        ,
        ⋅
        
          ⟩
          
            2
          
        
      
    
    {\displaystyle \langle \cdot ,\cdot \rangle _{2}}
  
 specify the inner product space.
</p><p>The space of square integrable functions is the <a href="/facts/Lp_space/gbad0Rv1">
  
    
      
        
          L
          
            p
          
        
      
    
    {\displaystyle L^{p}}
  
 space</a> in which 
  
    
      
        p
        =
        2.
      
    
    {\displaystyle p=2.}

</p>
<h2 id="examples">Examples</h2>
<p>The function 
  
    
      
        
          
            
              1
              
                x
                
                  n
                
              
            
          
        
        ,
      
    
    {\displaystyle {\tfrac {1}{x^{n}}},}
  
 defined on 
  
    
      
        (
        0
        ,
        1
        )
        ,
      
    
    {\displaystyle (0,1),}
  
 is in 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L^{2}}
  
 for 
  
    
      
        n
        <
        
          
            
              1
              2
            
          
        
      
    
    {\displaystyle n<{\tfrac {1}{2}}}
  
 but not for 
  
    
      
        n
        =
        
          
            
              1
              2
            
          
        
        .
      
    
    {\displaystyle n={\tfrac {1}{2}}.}
  
<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> 
The function 
  
    
      
        
          
            
              1
              x
            
          
        
        ,
      
    
    {\displaystyle {\tfrac {1}{x}},}
  
 defined on 
  
    
      
        [
        1
        ,
        ∞
        )
        ,
      
    
    {\displaystyle [1,\infty ),}
  
 is square-integrable.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p><p>Bounded functions, defined on 
  
    
      
        [
        0
        ,
        1
        ]
        ,
      
    
    {\displaystyle [0,1],}
  
 are square-integrable. These functions are also in 
  
    
      
        
          L
          
            p
          
        
        ,
      
    
    {\displaystyle L^{p},}
  
 for any value of 
  
    
      
        p
        .
      
    
    {\displaystyle p.}
  
<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h3>Non-examples</h3>
<p>The function 
  
    
      
        
          
            
              1
              x
            
          
        
        ,
      
    
    {\displaystyle {\tfrac {1}{x}},}
  
 defined on 
  
    
      
        [
        0
        ,
        1
        ]
        ,
      
    
    {\displaystyle [0,1],}
  
 where the value at 
  
    
      
        0
      
    
    {\displaystyle 0}
  
 is arbitrary. Furthermore, this function is not in 
  
    
      
        
          L
          
            p
          
        
      
    
    {\displaystyle L^{p}}
  
 for any value of 
  
    
      
        p
      
    
    {\displaystyle p}
  
 in 
  
    
      
        [
        1
        ,
        ∞
        )
        .
      
    
    {\displaystyle [1,\infty ).}
  
<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Inner_product_space/HoyjElEy">Inner product space</a></li>
<li><a href="/facts/Lp_space/gbad0Rv1">
  
    
      
        
          L
          
            p
          
        
      
    
    {\displaystyle L^{p}}
  
 space</a> – Function spaces generalizing finite-dimensional p norm spaces</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Todd, Rowland. "L^2-Function". MathWorld--A Wolfram Web Resource. <a href="http://mathworld.wolfram.com/L2-Function.html" target="_blank">http://mathworld.wolfram.com/L2-Function.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Giovanni Sansone (1991). Orthogonal Functions. Dover Publications. pp. 1–2. ISBN 978-0-486-66730-0. <a href="978-0-486-66730-0" target="_blank">978-0-486-66730-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Todd, Rowland. "L^2-Function". MathWorld--A Wolfram Web Resource. <a href="http://mathworld.wolfram.com/L2-Function.html" target="_blank">http://mathworld.wolfram.com/L2-Function.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>"Lp Functions" (PDF). Archived from the original (PDF) on 2020-10-24. Retrieved 2020-01-16. <a href="https://web.archive.org/web/20201024063542/http://faculty.bard.edu/belk/math461/LpFunctions.pdf" target="_blank">https://web.archive.org/web/20201024063542/http://faculty.bard.edu/belk/math461/LpFunctions.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"Lp Functions" (PDF). Archived from the original (PDF) on 2020-10-24. Retrieved 2020-01-16. <a href="https://web.archive.org/web/20201024063542/http://faculty.bard.edu/belk/math461/LpFunctions.pdf" target="_blank">https://web.archive.org/web/20201024063542/http://faculty.bard.edu/belk/math461/LpFunctions.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>"Lp Functions" (PDF). Archived from the original (PDF) on 2020-10-24. Retrieved 2020-01-16. <a href="https://web.archive.org/web/20201024063542/http://faculty.bard.edu/belk/math461/LpFunctions.pdf" target="_blank">https://web.archive.org/web/20201024063542/http://faculty.bard.edu/belk/math461/LpFunctions.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Square-integrable function open-in-new

Square-integrable function