Banach lattice

<h2 id="examples-and-constructions">Examples and constructions</h2>
<p>Banach lattices are extremely common in functional analysis, and "every known example [in 1948] of a Banach space [was] also a <a href="/facts/Vector_lattice/FX7Mwfxe">vector lattice</a>."<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>  In particular:
</p>
<ul><li>ℝ, together with its absolute value as a norm, is a Banach lattice.</li>
<li>Let X be a topological space, Y a Banach lattice and 𝒞(<i>X</i>,<i>Y</i>) the <a href="/facts/Space_of_continuous_bounded_functions/v5Yf5tlM">space of continuous bounded functions</a> from X to Y with norm 
  
    
      
        ‖
        f
        
          ‖
          
            ∞
          
        
        =
        
          sup
          
            x
            ∈
            X
          
        
        ‖
        f
        (
        x
        )
        
          ‖
          
            Y
          
        
        
          .
        
      
    
    {\displaystyle \|f\|_{\infty }=\sup _{x\in X}\|f(x)\|_{Y}{\text{.}}}
  
 Then 𝒞(<i>X</i>,<i>Y</i>) is a Banach lattice under the pointwise partial order: 
  
    
      
        
          f
          ≤
          g
        
        ⇔
        (
        ∀
        x
        ∈
        X
        )
        (
        f
        (
        x
        )
        ≤
        g
        (
        x
        )
        )
        
          .
        
      
    
    {\displaystyle {f\leq g}\Leftrightarrow (\forall x\in X)(f(x)\leq g(x)){\text{.}}}
  
</li></ul>
<p>Examples of non-lattice Banach spaces are now known; <a href="/facts/James_space/6urfAzPJ">James' space</a> is one such.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>  
</p>
<h2 id="properties">Properties</h2>
<p>The continuous dual space of a Banach lattice is equal to its <a href="/facts/Order_dual_(functional_analysis)/VgHtabVq">order dual</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> 
</p><p>Every Banach lattice admits a <a href="/facts/Approximate_identity/xjNY7eol">continuous approximation to the identity</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="abstract-l-spaces">Abstract (L)-spaces</h2>
<p>A Banach lattice satisfying the additional condition 
  
    
      
        
          f
          ,
          g
          ≥
          0
        
        ⇒
        ‖
        f
        +
        g
        ‖
        =
        ‖
        f
        ‖
        +
        ‖
        g
        ‖
      
    
    {\displaystyle {f,g\geq 0}\Rightarrow \|f+g\|=\|f\|+\|g\|}
  
 is called an abstract (L)-space.  Such spaces, under the assumption of separability, are isomorphic to closed sublattices of <i>L</i>1([0,1]).<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>  The classical <a href="/facts/Mean_ergodic_theorem/rkqCbIy0">mean ergodic theorem</a> and <a href="/facts/Poincar%25C3%25A9_recurrence/Uz2ja34X">Poincaré recurrence</a> generalize to abstract (L)-spaces.<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/Banach_space/sDEIk7EC">Banach space</a> – Normed vector space that is complete</li>
<li><a href="/facts/Normed_vector_lattice/zreeqV8e">Normed vector lattice</a></li>
<li><a href="/facts/Riesz_space/FX7Mwfxe">Riesz space</a> – Partially ordered vector space, ordered as a lattice</li>
<li><a href="/facts/Lattice_(order)/5ak6ufky">Lattice (order)</a> – Set whose pairs have minima and maxima</li></ul>
<h2 id="footnotes">Footnotes</h2>

<h2 id="bibliography">Bibliography</h2>
<ul><li>Abramovich, Yuri A.; Aliprantis, C. D. (2002). <i>An Invitation to Operator Theory</i>. Graduate Studies in Mathematics. Vol. 50. American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-2146-6.</li>
<li><a href="/facts/Garrett_Birkhoff/5jeI7Wmr">Birkhoff, Garrett</a> (1948). <a href="https://hdl.handle.net/2027/iau.31858027322886"><i>Lattice Theory</i></a>. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/2027%2Fiau.31858027322886">2027/iau.31858027322886</a> – via HathiTrust.</li>
<li>Narici, Lawrence; Beckenstein, Edward (2011). <i>Topological Vector Spaces</i>. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li>
<li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). <i>Topological Vector Spaces</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li></ul>

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

<ol>
<li id="fn:1"><p>Birkhoff 1948, p. 246. - Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. <a href="https://hdl.handle.net/2027/iau.31858027322886" target="_blank">https://hdl.handle.net/2027/iau.31858027322886</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Kania, Tomasz (12 April 2017).  Answer to "Banach space that is not a Banach lattice" (accessed 13 August 2022).  Mathematics StackExchange.  StackOverflow. <a href="https://math.stackexchange.com/a/2230649" target="_blank">https://math.stackexchange.com/a/2230649</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Schaefer & Wolff 1999, pp. 234–242. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Birkhoff 1948, p. 251. - Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. <a href="https://hdl.handle.net/2027/iau.31858027322886" target="_blank">https://hdl.handle.net/2027/iau.31858027322886</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Birkhoff 1948, pp. 250, 254. - Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. <a href="https://hdl.handle.net/2027/iau.31858027322886" target="_blank">https://hdl.handle.net/2027/iau.31858027322886</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Birkhoff 1948, pp. 269–271. - Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. <a href="https://hdl.handle.net/2027/iau.31858027322886" target="_blank">https://hdl.handle.net/2027/iau.31858027322886</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Banach lattice open-in-new

Banach lattice