Uniformly smooth space

<h2 id="properties">Properties</h2>
<ul><li>Every uniformly smooth <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> is <a href="/facts/Reflexive_space/g7Cjbw0S">reflexive</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li>
<li>A Banach space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is uniformly smooth if and only if its <a href="/facts/Dual_space/4Uw4Knz1">continuous dual</a> 
 
 
 
 
 X
 
 ∗
 
 
 
 
 {\displaystyle X^{*}}
 
 is <a href="/facts/Uniformly_convex_space/H9gGXdy7">uniformly convex</a> (and vice versa, via reflexivity).<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> The moduli of convexity and smoothness are linked by</li></ul>

ρ
          
            
              X
              
                ∗
              
            
          
        
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        =
        sup
        {
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          δ
          
            X
          
        
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        ε
        ∈
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        ,
        
        t
        ≥
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    {\displaystyle \rho _{X^{*}}(t)=\sup\{t\varepsilon /2-\delta _{X}(\varepsilon ):\varepsilon \in [0,2]\},\quad t\geq 0,}

and the maximal convex function majorated by the modulus of convexity δX is given by<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

δ
                ~
              
            
          
          
            X
          
        
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          ρ
          
            
              X
              
                ∗
              
            
          
        
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        t
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        :
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        ≥
        0
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        .
      
    
    {\displaystyle {\tilde {\delta }}_{X}(\varepsilon )=\sup\{\varepsilon t/2-\rho _{X^{*}}(t):t\geq 0\}.}

Furthermore,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

δ
          
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        ≤
        
          
            
              
                δ
                ~
              
            
          
          
            X
          
        
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          δ
          
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    {\displaystyle \delta _{X}(\varepsilon /2)\leq {\tilde {\delta }}_{X}(\varepsilon )\leq \delta _{X}(\varepsilon ),\quad \varepsilon \in [0,2].}

<ul><li>A Banach space is uniformly smooth if and only if the limit</li></ul>

lim
          
            t
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              ‖
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    {\displaystyle \lim _{t\to 0}{\frac {\|x+ty\|-\|x\|}{t}}}

exists uniformly for all 
 
 
 
 x
 ,
 y
 ∈
 
 S
 
 X
 
 
 
 
 {\displaystyle x,y\in S_{X}}
 
 (where 
 
 
 
 
 S
 
 X
 
 
 
 
 {\displaystyle S_{X}}
 
 denotes the <a href="/facts/Unit_sphere/5UeoTcug">unit sphere</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
).
<ul><li>When 1 < p < ∞, the <a href="/facts/Lp_space/gbad0Rv1">Lp-spaces</a> are uniformly smooth (and uniformly convex).</li></ul>
<a href="/facts/Per_Enflo/YneYxtRn">Enflo</a> proved<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> 
that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of <a href="/facts/Reflexive_space/g7Cjbw0S">super-reflexive</a> Banach spaces, introduced by <a href="/facts/Robert_C._James/lfIrQSPX">Robert C. James</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> 
As a space is super-reflexive if and only if its dual is super-reflexive, it follows that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of spaces that admit an equivalent uniformly smooth norm. The <a href="/facts/Gilles_Pisier/GzRQq0yx">Pisier</a> renorming theorem<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> 
states that a super-reflexive space X admits an equivalent uniformly smooth norm for which the modulus of smoothness ρX satisfies, for some constant C and some p > 1

ρ
          
            X
          
        
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        t
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        ≤
        C
        
        
          t
          
            p
          
        
        ,
        
        t
        >
        0.
      
    
    {\displaystyle \rho _{X}(t)\leq C\,t^{p},\quad t>0.}

It follows that every super-reflexive space Y admits an equivalent uniformly convex norm for which the <a href="/facts/Modulus_and_characteristic_of_convexity/p2K6tKRP">modulus of convexity</a> satisfies, for some constant c > 0 and some positive real q

δ
          
            Y
          
        
        (
        ε
        )
        ≥
        c
        
        
          ε
          
            q
          
        
        ,
        
        ε
        ∈
        [
        0
        ,
        2
        ]
        .
      
    
    {\displaystyle \delta _{Y}(\varepsilon )\geq c\,\varepsilon ^{q},\quad \varepsilon \in [0,2].}

If a normed space admits two equivalent norms, one uniformly convex and one uniformly smooth, the Asplund averaging technique<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> produces another equivalent norm that is both uniformly convex and uniformly smooth.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Uniformly_convex_space/H9gGXdy7">Uniformly convex space</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Diestel, Joseph (1984), <a href="https://archive.org/details/sequencesseriesi0000dies">Sequences and series in Banach spaces</a>, Graduate Texts in Mathematics, vol. 92, New York: Springer-Verlag, pp. xii+261, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90859-5</li>
<li>Itô, Kiyosi (1993), Encyclopedic Dictionary of Mathematics, Volume 1, MIT Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-262-59020-4 <a href="https://books.google.com/books?id=WHjO9K6xEm4C">[1]</a></li>
<li><a href="/facts/Joram_Lindenstrauss/LW2BeVTd">Lindenstrauss, Joram</a>; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-08888-1.</li></ul>

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

<ol>
<li id="fn:1">see Definition 1.e.1, p. 59 in Lindenstrauss & Tzafriri (1979). - Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Proposition 1.e.3, p. 61 in Lindenstrauss & Tzafriri (1979). - Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Proposition 1.e.2, p. 61 in Lindenstrauss & Tzafriri (1979). - Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Proposition 1.e.6, p. 65 in Lindenstrauss & Tzafriri (1979). - Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Lemma 1.e.7 and 1.e.8, p. 66 in Lindenstrauss & Tzafriri (1979). - Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Enflo, Per (1973), "Banach spaces which can be given an equivalent uniformly convex norm", Israel Journal of Mathematics, 13 (3–4): 281–288, doi:10.1007/BF02762802 <a href="/wiki/Per_Enflo" target="_blank">/wiki/Per_Enflo</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">James, Robert C. (1972), "Super-reflexive Banach spaces", Canadian Journal of Mathematics, 24 (5): 896–904, doi:10.4153/CJM-1972-089-7 <a href="/wiki/Robert_C._James" target="_blank">/wiki/Robert_C._James</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel Journal of Mathematics, 20 (3–4): 326–350, doi:10.1007/BF02760337 <a href="/wiki/Israel_Journal_of_Mathematics" target="_blank">/wiki/Israel_Journal_of_Mathematics</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Asplund, Edgar (1967), "Averaged norms", Israel Journal of Mathematics, 5 (4): 227–233, doi:10.1007/BF02771611 <a href="/wiki/Israel_Journal_of_Mathematics" target="_blank">/wiki/Israel_Journal_of_Mathematics</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Uniformly smooth space open-in-new

Uniformly smooth space