Unconditional convergence

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a>. Let 
  
    
      
        I
      
    
    {\displaystyle I}
  
 be an <a href="/facts/Index_set/Jta7dIan">index set</a> and 
  
    
      
        
          x
          
            i
          
        
        ∈
        X
      
    
    {\displaystyle x_{i}\in X}
  
 for all 
  
    
      
        i
        ∈
        I
        .
      
    
    {\displaystyle i\in I.}

</p><p>The series 
  
    
      
        
          
            ∑
            
              i
              ∈
              I
            
          
          
            x
            
              i
            
          
        
      
    
    {\displaystyle \textstyle \sum _{i\in I}x_{i}}
  
 is called unconditionally convergent to  
  
    
      
        x
        ∈
        X
        ,
      
    
    {\displaystyle x\in X,}
  
 if
</p>
<ul><li>the indexing set 
  
    
      
        
          I
          
            0
          
        
        :=
        
          {
          
            i
            ∈
            I
            :
            
              x
              
                i
              
            
            ≠
            0
          
          }
        
      
    
    {\displaystyle I_{0}:=\left\{i\in I:x_{i}\neq 0\right\}}
  
 is <a href="/facts/Countable/Wj4DmWPv">countable</a>, and</li>
<li>for every <a href="/facts/Permutation/JSw9ukmv">permutation</a> (<a href="/facts/Bijection/j4gTuTmW">bijection</a>) 
  
    
      
        σ
        :
        
          I
          
            0
          
        
        →
        
          I
          
            0
          
        
      
    
    {\displaystyle \sigma :I_{0}\to I_{0}}
  
 of 
  
    
      
        
          I
          
            0
          
        
        =
        
          
            {
            
              i
              
                k
              
            
            }
          
          
            k
            =
            1
          
          
            ∞
          
        
      
    
    {\displaystyle I_{0}=\left\{i_{k}\right\}_{k=1}^{\infty }}
  
 the following relation holds: 
  
    
      
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          x
          
            σ
            
              (
              
                i
                
                  k
                
              
              )
            
          
        
        =
        x
        .
      
    
    {\displaystyle \sum _{k=1}^{\infty }x_{\sigma \left(i_{k}\right)}=x.}
  
</li></ul>
<h2 id="alternative-definition">Alternative definition</h2>
<p>Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence 
  
    
      
        
          
            (
            
              ε
              
                n
              
            
            )
          
          
            n
            =
            1
          
          
            ∞
          
        
        ,
      
    
    {\displaystyle \left(\varepsilon _{n}\right)_{n=1}^{\infty },}
  
 with 
  
    
      
        
          ε
          
            n
          
        
        ∈
        {
        −
        1
        ,
        +
        1
        }
        ,
      
    
    {\displaystyle \varepsilon _{n}\in \{-1,+1\},}
  
 the series

∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          ε
          
            n
          
        
        
          x
          
            n
          
        
      
    
    {\displaystyle \sum _{n=1}^{\infty }\varepsilon _{n}x_{n}}

converges.
</p><p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>, every <a href="/facts/Absolute_convergence/BXz9kBJ5">absolutely convergent</a> series is unconditionally convergent, but the <a href="/facts/Converse_(logic)/VhlUrxxx">converse</a> implication does not hold in general. Indeed, if 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is an infinite-dimensional Banach space, then by <a href="/facts/Absolute_convergence/BXz9kBJ5">Dvoretzky–Rogers theorem</a> there always exists an unconditionally convergent series in this space that is not absolutely convergent. However, when 
  
    
      
        X
        =
        
          
            R
          
          
            n
          
        
        ,
      
    
    {\displaystyle X=\mathbb {R} ^{n},}
  
 by the <a href="/facts/Riemann_series_theorem/YHsT2BEU">Riemann series theorem</a>, the series 
  
    
      
        
          ∑
          
            n
          
        
        
          x
          
            n
          
        
      
    
    {\textstyle \sum _{n}x_{n}}
  
 is unconditionally convergent if and only if it is absolutely convergent.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Absolute_convergence/BXz9kBJ5">Absolute convergence</a> – Mode of convergence of an infinite series</li>
<li><a href="/facts/Modes_of_convergence_(annotated_index)/WEEA7Ctm">Modes of convergence (annotated index)</a> – Annotated index of various modes of convergence</li>
<li><a href="/facts/Absolute_convergence/BXz9kBJ5">Rearrangements and unconditional convergence/Dvoretzky–Rogers theorem</a> – Mode of convergence of an infinite series</li>
<li><a href="/facts/Riemann_series_theorem/YHsT2BEU">Riemann series theorem</a> – Unconditionally convergent series converge absolutely</li></ul>

<ul><li>Ch. Heil: <a href="http://www.math.gatech.edu/~heil/papers/bases.pdf">A Basis Theory Primer</a></li>
<li>Knopp, Konrad (1956). <a href="https://archive.org/details/infinitesequence0000knop"><i>Infinite Sequences and Series</i></a>. Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780486601533.</li>
<li>Knopp, Konrad (1990). <i>Theory and Application of Infinite Series</i>. Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780486661650.</li>
<li>Wojtaszczyk, P. (1996). <i>Banach spaces for analysts</i>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780521566759.</li></ul>

<p><i>This article incorporates material from <a href="https://planetmath.org/unconditionalconvergence">Unconditional convergence</a> on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i>
</p>

Unconditional convergence open-in-new

Unconditional convergence