Delta-convergence

<h2 id="definition">Definition</h2>
<p>A sequence 
  
    
      
        (
        
          x
          
            k
          
        
        )
      
    
    {\displaystyle (x_{k})}
  
 in a metric space 
  
    
      
        (
        X
        ,
        d
        )
      
    
    {\displaystyle (X,d)}
  
 is said to be  Δ-convergent to 
  
    
      
        x
        ∈
        X
      
    
    {\displaystyle x\in X}
  
 if for every 
  
    
      
        y
        ∈
        X
      
    
    {\displaystyle y\in X}
  
, 
  
    
      
        lim sup
        (
        d
        (
        
          x
          
            k
          
        
        ,
        x
        )
        −
        d
        (
        
          x
          
            k
          
        
        ,
        y
        )
        )
        ≤
        0
      
    
    {\displaystyle \limsup(d(x_{k},x)-d(x_{k},y))\leq 0}
  
.
</p>
<h2 id="characterization-in-banach-spaces">Characterization in Banach spaces</h2>
<p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a <a href="/facts/Uniformly_convex_space/H9gGXdy7">uniformly convex</a> and <a href="/facts/Uniformly_smooth_space/715KDNln">uniformly smooth</a> Banach space, with the duality mapping  
  
    
      
        x
        ↦
        
          x
          
            ∗
          
        
      
    
    {\displaystyle x\mapsto x^{*}}
  
 given by  
  
    
      
        ‖
        x
        ‖
        =
        ‖
        
          x
          
            ∗
          
        
        ‖
      
    
    {\displaystyle \|x\|=\|x^{*}\|}
  
,  
  
    
      
        ⟨
        
          x
          
            ∗
          
        
        ,
        x
        ⟩
        =
        ‖
        x
        
          ‖
          
            2
          
        
      
    
    {\displaystyle \langle x^{*},x\rangle =\|x\|^{2}}
  
, then a sequence 
  
    
      
        (
        
          x
          
            k
          
        
        )
        ⊂
        X
      
    
    {\displaystyle (x_{k})\subset X}
  
 is Delta-convergent to 
  
    
      
        x
      
    
    {\displaystyle x}
  
 if and only if 
  
    
      
        (
        
          x
          
            k
          
        
        −
        x
        
          )
          
            ∗
          
        
      
    
    {\displaystyle (x_{k}-x)^{*}}
  
 converges to zero weakly in the dual space 
  
    
      
        
          X
          
            ∗
          
        
      
    
    {\displaystyle X^{*}}
  
 (see <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>). In particular, Delta-convergence and weak convergence coincide if 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a Hilbert space.
</p>
<h3>Opial property</h3>
<p>Coincidence of weak convergence and Delta-convergence is equivalent, for uniformly convex Banach spaces, to the well-known 
<a href="/facts/Opial_property/jd8gNNch">Opial property</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="delta-compactness-theorem">Delta-compactness theorem</h2>
<p>The Delta-compactness theorem of T. C. Lim<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> states that if 
  
    
      
        (
        X
        ,
        d
        )
      
    
    {\displaystyle (X,d)}
  
 is an <i>asymptotically complete</i> metric space, then every bounded sequence in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 has a Delta-convergent subsequence.
</p><p>The Delta-compactness theorem is similar to the <a href="/facts/Banach%25E2%2580%2593Alaoglu_theorem/NILk5nDm">Banach–Alaoglu theorem</a> for weak convergence but, unlike the Banach-Alaoglu theorem (in the non-separable case) its proof does not depend on the Axiom of Choice.
</p>
<h3>Asymptotic center and asymptotic completeness</h3>
<p>An <i>asymptotic center</i> of a sequence 
  
    
      
        (
        
          x
          
            k
          
        
        
          )
          
            k
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (x_{k})_{k\in \mathbb {N} }}
  
, if it exists, is a limit of the <a href="/facts/Chebyshev_center/vwuEKFgH">Chebyshev centers</a> 
  
    
      
        
          c
          
            n
          
        
      
    
    {\displaystyle c_{n}}
  
  for truncated sequences 
  
    
      
        (
        
          x
          
            k
          
        
        
          )
          
            k
            ≥
            n
          
        
      
    
    {\displaystyle (x_{k})_{k\geq n}}
  
. A metric space is called <i>asymptotically complete</i>, if any bounded sequence in it has an asymptotic center.
</p>
<h3>Uniform convexity as sufficient condition of asymptotic completeness</h3>
<p>Condition of asymptotic completeness in the Delta-compactness theorem is satisfied by uniformly convex Banach spaces, and more generally, by uniformly rotund metric spaces as defined by J. Staples.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h2 id="further-reading">Further reading</h2>
<ul><li>William Kirk, Naseer Shahzad, Fixed point theory in distance spaces. Springer, Cham, 2014. xii+173 pp.</li>
<li>G. Devillanova, S. Solimini, C. Tintarev, On weak convergence in metric spaces, Nonlinear Analysis and Optimization (B. S. Mordukhovich, S. Reich, A. J. Zaslavski, Editors), 43–64, Contemporary Mathematics 659, AMS, Providence, RI, 2016.</li></ul>

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

<ol>
<li id="fn:1"><p>T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179–182. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>T. Kuczumow, An almost convergence and its applications, Ann. Univ. Mariae Curie-Sklodowska Sect. A  32 (1978), 79–88. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>S. Solimini, C. Tintarev, Concentration analysis in Banach spaces, Comm. Contemp. Math. 2015, DOI 10.1142/S0219199715500388 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>S. Solimini, C. Tintarev, Concentration analysis in Banach spaces, Comm. Contemp. Math. 2015, DOI 10.1142/S0219199715500388 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179–182. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>J. Staples, Fixed point theorems in uniformly rotund metric spaces, Bull. Austral. Math. Soc. 14 (1976), 181–192. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Delta-convergence open-in-new

Delta-convergence