F-space

<h2 id="examples">Examples</h2>
<p>All <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a> and <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a> are F-spaces. In particular, a Banach space is an F-space with an additional requirement that 
  
    
      
        d
        (
        a
        x
        ,
        0
        )
        =
        
          |
        
        a
        
          |
        
        d
        (
        x
        ,
        0
        )
        .
      
    
    {\displaystyle d(ax,0)=|a|d(x,0).}
  
<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>The <a href="/facts/Lp_space/gbad0Rv1">Lp spaces</a> can be made into F-spaces for all 
  
    
      
        p
        ≥
        0
      
    
    {\displaystyle p\geq 0}
  
 and for 
  
    
      
        p
        ≥
        1
      
    
    {\displaystyle p\geq 1}
  
 they can be made into locally convex and thus Fréchet spaces and even Banach spaces.
</p>
<h3>Example 1</h3>
<p>
  
    
      
        
          L
          
            
              1
              2
            
          
        
        [
        0
        ,
        
        1
        ]
      
    
    {\displaystyle L^{\frac {1}{2}}[0,\,1]}
  
 is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial <a href="/facts/Dual_space/4Uw4Knz1">dual space</a>.
</p>
<h3>Example 2</h3>
<p>Let 
  
    
      
        
          W
          
            p
          
        
        (
        
          D
        
        )
      
    
    {\displaystyle W_{p}(\mathbb {D} )}
  
 be the space of all complex valued <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a>

f
        (
        z
        )
        =
        
          ∑
          
            n
            ≥
            0
          
        
        
          a
          
            n
          
        
        
          z
          
            n
          
        
      
    
    {\displaystyle f(z)=\sum _{n\geq 0}a_{n}z^{n}}

on the unit disc 
  
    
      
        
          D
        
      
    
    {\displaystyle \mathbb {D} }
  
 such that

∑
          
            n
          
        
        
          
            |
            
              a
              
                n
              
            
            |
          
          
            p
          
        
        <
        ∞
      
    
    {\displaystyle \sum _{n}\left|a_{n}\right|^{p}<\infty }

then for 
  
    
      
        0
        <
        p
        <
        1
        ,
      
    
    {\displaystyle 0<p<1,}
  
 
  
    
      
        
          W
          
            p
          
        
        (
        
          D
        
        )
      
    
    {\displaystyle W_{p}(\mathbb {D} )}
  
 are F-spaces under the <a href="/facts/P-norm/gbad0Rv1">p-norm</a>:

‖
        f
        
          ‖
          
            p
          
        
        =
        
          ∑
          
            n
          
        
        
          
            |
            
              a
              
                n
              
            
            |
          
          
            p
          
        
        
        (
        0
        <
        p
        <
        1
        )
        .
      
    
    {\displaystyle \|f\|_{p}=\sum _{n}\left|a_{n}\right|^{p}\qquad (0<p<1).}

</p><p>In fact, 
  
    
      
        
          W
          
            p
          
        
      
    
    {\displaystyle W_{p}}
  
 is a <a href="/facts/Quasi-Banach_algebra/ledoUdJW">quasi-Banach algebra</a>. Moreover, for any 
  
    
      
        ζ
      
    
    {\displaystyle \zeta }
  
 with 
  
    
      
        
          |
        
        ζ
        
          |
        
        ≤
        1
      
    
    {\displaystyle |\zeta |\leq 1}
  
 the map 
  
    
      
        f
        ↦
        f
        (
        ζ
        )
      
    
    {\displaystyle f\mapsto f(\zeta )}
  
 is a bounded linear (multiplicative functional) on 
  
    
      
        
          W
          
            p
          
        
        (
        
          D
        
        )
        .
      
    
    {\displaystyle W_{p}(\mathbb {D} ).}

</p>
<h2 id="sufficient-conditions">Sufficient conditions</h2>

<p>Theorem<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> (Klee (1952))—Let 
  
    
      
        d
      
    
    {\displaystyle d}
  
 be any<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> metric on a vector space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 such that the topology 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 induced by 
  
    
      
        d
      
    
    {\displaystyle d}
  
 on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 makes 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 into a topological vector space. If 
  
    
      
        (
        X
        ,
        d
        )
      
    
    {\displaystyle (X,d)}
  
 is a complete metric space then 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 is a <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete topological vector space</a>.
</p>

<h2 id="related-properties">Related properties</h2>
<p>The <a href="/facts/Open_mapping_theorem_(functional_analysis)/nFChD5j5">open mapping theorem</a> implies that if 
  
    
      
        τ
        
           and 
        
        
          τ
          
            2
          
        
      
    
    {\displaystyle \tau {\text{ and }}\tau _{2}}
  
 are topologies on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 that make both 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 and 
  
    
      
        
          (
          
            X
            ,
            
              τ
              
                2
              
            
          
          )
        
      
    
    {\displaystyle \left(X,\tau _{2}\right)}
  
 into <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete</a> <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable topological vector spaces</a> (for example, Banach or <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a>) and if one topology is <a href="/facts/Comparison_of_topologies/5Q9F2fjf">finer or coarser</a> than the other then they must be equal (that is, if 
  
    
      
        τ
        ⊆
        
          τ
          
            2
          
        
        
           or 
        
        
          τ
          
            2
          
        
        ⊆
        τ
        
           then 
        
        τ
        =
        
          τ
          
            2
          
        
      
    
    {\displaystyle \tau \subseteq \tau _{2}{\text{ or }}\tau _{2}\subseteq \tau {\text{ then }}\tau =\tau _{2}}
  
).<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<ul><li>A linear almost continuous map into an F-space whose graph is closed is continuous.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li>
<li>A linear <a href="/facts/Almost_open_map/ta3sk8AY">almost open</a> map into an F-space whose graph is closed is necessarily an <a href="/facts/Open_map/abHUKX7l">open map</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li>
<li>A linear continuous <a href="/facts/Almost_open_map/ta3sk8AY">almost open</a> map from an F-space is necessarily an <a href="/facts/Open_map/abHUKX7l">open map</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li>
<li>A linear continuous almost open map from an F-space whose image is of the <a href="/facts/Second_category/AxTk8frw">second category</a> in the codomain is necessarily a <a href="/facts/Surjection/KezhLpCb">surjective</a> <a href="/facts/Open_map/abHUKX7l">open map</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li></ul>
<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/Barreled_space/r2H58TN3">Barreled space</a> – Type of topological vector spacePages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Countably_quasi-barrelled_space/q1SuqeYk">Countably quasi-barrelled space</a></li>
<li><a href="/facts/Complete_metric_space/lsckmU8G">Complete metric space</a> – Metric geometry</li>
<li><a href="/facts/Complete_topological_vector_space/e6qtS76j">Complete topological vector space</a> – A TVS where points that get progressively closer to each other will always converge to a point</li>
<li><a href="/facts/DF-space/csjJmkad">DF-space</a> – class of special local-convex spacePages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet space</a> – Locally convex topological vector space that is also a complete metric space</li>
<li><a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> – Type of topological vector space</li>
<li><a href="/facts/K-space_(functional_analysis)/m7ImvGjg">K-space (functional analysis)</a></li>
<li><a href="/facts/LB-space/Dg19Fq32">LB-space</a></li>
<li><a href="/facts/LF-space/voQHEeGI">LF-space</a> – Topological vector space</li>
<li><a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">Metrizable topological vector space</a> – A topological vector space whose topology can be defined by a metric</li>
<li><a href="/facts/Nuclear_space/4FLPWdfp">Nuclear space</a> – A generalization of finite-dimensional Euclidean spaces different from Hilbert spaces</li>
<li><a href="/facts/Projective_tensor_product/lv5X7Ps7">Projective tensor product</a> – tensor product defined on two topological vector spacesPages displaying wikidata descriptions as a fallback</li></ul>

<h2 id="notes">Notes</h2>

<h2 id="sources">Sources</h2>
<ul><li>Husain, Taqdir; Khaleelulla, S. M. (1978). <i>Barrelledness in Topological and Ordered Vector Spaces</i>. <a href="/facts/Lecture_Notes_in_Mathematics/5DmSFMhJ">Lecture Notes in Mathematics</a>. Vol. 692. Berlin, New York, Heidelberg: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-09096-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/4493665">4493665</a>.</li>
<li>Khaleelulla, S. M. (1982). <i>Counterexamples in Topological Vector Spaces</i>. <a href="/facts/Lecture_Notes_in_Mathematics/5DmSFMhJ">Lecture Notes in Mathematics</a>. Vol. 936. Berlin, Heidelberg, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-11565-6. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/8588370">8588370</a>.</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/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1966). <i>Real & Complex Analysis</i>. McGraw-Hill. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-054234-1.</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1991). <a href="https://archive.org/details/functionalanalys00rudi"><i>Functional Analysis</i></a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/facts/McGraw-Hill_Science%2fEngineering%2fMath/srSyQSsY">McGraw-Hill Science/Engineering/Math</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054236-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21163277">21163277</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>
<li><a href="/facts/Eric_Schechter/cApyLyxB">Schechter, Eric</a> (1996). <i>Handbook of Analysis and Its Foundations</i>. San Diego, CA: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-622760-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/175294365">175294365</a>.</li>
<li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li></ul>

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

<ol>
<li id="fn:1"><p>Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Schaefer & Wolff 1999, p. 35. - 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:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4. <a href="https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf" target="_blank">https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Not assume to be translation-invariant. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Trèves 2006, pp. 166–173. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Husain & Khaleelulla 1978, p. 14. - Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. <a href="https://search.worldcat.org/oclc/4493665" target="_blank">https://search.worldcat.org/oclc/4493665</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Husain & Khaleelulla 1978, p. 14. - Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. <a href="https://search.worldcat.org/oclc/4493665" target="_blank">https://search.worldcat.org/oclc/4493665</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Husain & Khaleelulla 1978, p. 15. - Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. <a href="https://search.worldcat.org/oclc/4493665" target="_blank">https://search.worldcat.org/oclc/4493665</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Husain & Khaleelulla 1978, p. 14. - Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. <a href="https://search.worldcat.org/oclc/4493665" target="_blank">https://search.worldcat.org/oclc/4493665</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
</ol>

F-space open-in-new

F-space