Bochner space

<h2 id="definition">Definition</h2>
<p>Given a <a href="/facts/Measure_space/w9Dx4KY1">measure space</a> 
  
    
      
        (
        T
        ,
        Σ
        ;
        μ
        )
        ,
      
    
    {\displaystyle (T,\Sigma ;\mu ),}
  
 a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> 
  
    
      
        
          (
          
            X
            ,
            ‖
            
            ⋅
            
            
              ‖
              
                X
              
            
          
          )
        
      
    
    {\displaystyle \left(X,\|\,\cdot \,\|_{X}\right)}
  
 and 
  
    
      
        1
        ≤
        p
        ≤
        ∞
        ,
      
    
    {\displaystyle 1\leq p\leq \infty ,}
  
 the Bochner space 
  
    
      
        
          L
          
            p
          
        
        (
        T
        ;
        X
        )
      
    
    {\displaystyle L^{p}(T;X)}
  
 is defined to be the <a href="/facts/Kolmogorov_quotient/dk7qNoRn">Kolmogorov quotient</a> (by equality <a href="/facts/Almost_everywhere/1quRFtJE">almost everywhere</a>) of the space of all <a href="/facts/Bochner_measurable/HxXFeAgt">Bochner measurable</a> functions 
  
    
      
        u
        :
        T
        →
        X
      
    
    {\displaystyle u:T\to X}
  
 such that the corresponding norm is finite:

‖
        u
        
          ‖
          
            
              L
              
                p
              
            
            (
            T
            ;
            X
            )
          
        
        :=
        
          
            (
            
              
                ∫
                
                  T
                
              
              ‖
              u
              (
              t
              )
              
                ‖
                
                  X
                
                
                  p
                
              
              
              
                d
              
              μ
              (
              t
              )
            
            )
          
          
            1
            
              /
            
            p
          
        
        <
        +
        ∞
        
          
             for 
          
        
        1
        ≤
        p
        <
        ∞
        ,
      
    
    {\displaystyle \|u\|_{L^{p}(T;X)}:=\left(\int _{T}\|u(t)\|_{X}^{p}\,\mathrm {d} \mu (t)\right)^{1/p}<+\infty {\mbox{ for }}1\leq p<\infty ,}

‖
        u
        
          ‖
          
            
              L
              
                ∞
              
            
            (
            T
            ;
            X
            )
          
        
        :=
        
          
            e
            s
            s
            
            s
            u
            p
          
          
            t
            ∈
            T
          
        
        ‖
        u
        (
        t
        )
        
          ‖
          
            X
          
        
        <
        +
        ∞
        .
      
    
    {\displaystyle \|u\|_{L^{\infty }(T;X)}:=\mathrm {ess\,sup} _{t\in T}\|u(t)\|_{X}<+\infty .}

</p><p>In other words, as is usual in the study of 
  
    
      
        
          L
          
            p
          
        
      
    
    {\displaystyle L^{p}}
  
 spaces, 
  
    
      
        
          L
          
            p
          
        
        (
        T
        ;
        X
        )
      
    
    {\displaystyle L^{p}(T;X)}
  
 is a space of <a href="/facts/Equivalence_class/1d2sh0TB">equivalence classes</a> of functions, where two functions are defined to be equivalent if they are equal everywhere except upon a 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
-<a href="/facts/Measure_zero/kMudL8Mn">measure zero</a> subset of 
  
    
      
        T
        .
      
    
    {\displaystyle T.}
  
 As is also usual in the study of such spaces, it is usual to <a href="/facts/Abuse_of_notation/D4W6n85F">abuse notation</a> and speak of a "function" in 
  
    
      
        
          L
          
            p
          
        
        (
        T
        ;
        X
        )
      
    
    {\displaystyle L^{p}(T;X)}
  
 rather than an equivalence class (which would be more technically correct).
</p>
<h2 id="applications">Applications</h2>
<p>Bochner spaces are often used in the <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> approach to the study of <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a> that depend on time, e.g. the <a href="/facts/Heat_equation/2JPPr5GG">heat equation</a>: if the temperature 
  
    
      
        g
        (
        t
        ,
        x
        )
      
    
    {\displaystyle g(t,x)}
  
 is a scalar function of time and space, one can write 
  
    
      
        (
        f
        (
        t
        )
        )
        (
        x
        )
        :=
        g
        (
        t
        ,
        x
        )
      
    
    {\displaystyle (f(t))(x):=g(t,x)}
  
 to make 
  
    
      
        f
      
    
    {\displaystyle f}
  
 a family 
  
    
      
        f
        (
        t
        )
      
    
    {\displaystyle f(t)}
  
 (parametrized by time) of functions of space, possibly in some Bochner space.
</p>
<h3>Application to PDE theory</h3>
<p>Very often, the space 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is an <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> of time over which we wish to solve some partial differential equation, and 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 will be one-dimensional <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a>. The idea is to regard a function of time and space as a collection of functions of space, this collection being parametrized by time. For example, in the solution of the heat equation on a region 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
 in 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 and an interval of time 
  
    
      
        [
        0
        ,
        T
        ]
        ,
      
    
    {\displaystyle [0,T],}
  
 one seeks solutions

u
        ∈
        
          L
          
            2
          
        
        
          (
          
            [
            0
            ,
            T
            ]
            ;
            
              H
              
                0
              
              
                1
              
            
            (
            Ω
            )
          
          )
        
      
    
    {\displaystyle u\in L^{2}\left([0,T];H_{0}^{1}(\Omega )\right)}

with time derivative

∂
              u
            
            
              ∂
              t
            
          
        
        ∈
        
          L
          
            2
          
        
        
          (
          
            [
            0
            ,
            T
            ]
            ;
            
              H
              
                −
                1
              
            
            (
            Ω
            )
          
          )
        
        .
      
    
    {\displaystyle {\frac {\partial u}{\partial t}}\in L^{2}\left([0,T];H^{-1}(\Omega )\right).}

Here 
  
    
      
        
          H
          
            0
          
          
            1
          
        
        (
        Ω
        )
      
    
    {\displaystyle H_{0}^{1}(\Omega )}
  
 denotes the <a href="/facts/Sobolev_space/34qxWCyr">Sobolev</a> <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> of once-<a href="/facts/Weak_derivative/oehSmSMg">weakly differentiable</a> functions with first weak derivative in 
  
    
      
        
          L
          
            2
          
        
        (
        Ω
        )
      
    
    {\displaystyle L^{2}(\Omega )}
  
 that vanish at the <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Support_(mathematics)/BTuMGbsb">support</a> in Ω); 
  
    
      
        
          H
          
            −
            1
          
        
        (
        Ω
        )
      
    
    {\displaystyle H^{-1}(\Omega )}
  
 denotes the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> of 
  
    
      
        
          H
          
            0
          
          
            1
          
        
        (
        Ω
        )
        .
      
    
    {\displaystyle H_{0}^{1}(\Omega ).}

</p><p>(The "<a href="/facts/Partial_derivative/JWHsBkAu">partial derivative</a>" with respect to time 
  
    
      
        t
      
    
    {\displaystyle t}
  
 above is actually a <a href="/facts/Total_derivative/cPwsm9Wf">total derivative</a>, since the use of Bochner spaces removes the space-dependence.)
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bochner_integral/RpzcTu9n">Bochner integral</a> – Concept in mathematics</li>
<li><a href="/facts/Bochner_measurable_function/HxXFeAgt">Bochner measurable function</a></li>
<li><a href="/facts/Vector_measure/pQXUI3mv">Vector measure</a></li>
<li><a href="/facts/Vector-valued_functions/qxu8aapt">Vector-valued functions</a> – Function valued in a vector space; typically a real or complex onePages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Weakly_measurable_function/cAFxsUvL">Weakly measurable function</a></li></ul>

<ul><li>Evans, Lawrence C. (1998). <i>Partial differential equations</i>. Providence, RI: American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0772-2.</li></ul>

Bochner space open-in-new

Bochner space