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A [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r0805701.png" /> that coincides under the canonical imbedding with its second dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r0805702.png" /> (cf. [[Adjoint space|Adjoint space]]). More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r0805703.png" /> be the space dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r0805704.png" />, i.e. the set of all continuous linear functionals defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r0805705.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r0805706.png" /> is the value of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r0805707.png" /> on an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r0805708.png" />, then with a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r0805709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057010.png" /> running through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057011.png" />, the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057012.png" /> defines a linear functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057013.png" />, that is, an element of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057014.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057015.png" /> be the set of such functionals. The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057016.png" /> is an isomorphism which does not change the norm: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057018.png" />, then the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057019.png" /> is called reflexive. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057022.png" />, are reflexive, and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057023.png" /> is not reflexive.
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A [[Banach space]] $X$ that coincides under the canonical imbedding with its second dual $X^{{*}{*}}$ (cf. [[Adjoint space]]). More precisely, let $X^{*}$ be the space dual to $X$, i.e. the set of all continuous linear functionals defined on $X$. If $(x,f)$ is the value of the functional $f \in X^{*}$ on an element $x \in X$, then with a fixed $x$ and $f$ running through $X^{*}$, the formula $(x,f) = \mathcal{F}_x(f)$ defines a linear functional on $X^{*}$, that is, an element of the space $X^{{*}{*}}$. Let $\tilde X \subseteq X^{{*}{*}}$ be the set of such functionals. The correspondence $x \mapsto \mathcal{F}_x$ is an isomorphism which does not change the norm: $\Vert x \Vert = \Vert \mathcal{F}_x \Vert$. If $\tilde X = X^{{*}{*}}$, then the space $X$ is called reflexive. The spaces $\ell_p$ and $L_p(a,b)$, $p>1$, are reflexive, and the space $C[a,b]$ is not reflexive.
  
A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057024.png" /> is reflexive if and only if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057025.png" /> is reflexive. Another criteria of reflexivity of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080570/r08057026.png" /> is weak compactness (cf. [[Weak topology|Weak topology]]) of the unit ball of this space.
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A space $X$ is reflexive if and only if the space $X^{*}$ is reflexive. Another criteria of reflexivity of a Banach space $X$ is weak compactness (cf. [[Weak topology]]) of the unit ball of this space.
  
 
A reflexive space is weakly complete and a closed subspace of a reflexive space is reflexive.
 
A reflexive space is weakly complete and a closed subspace of a reflexive space is reflexive.
  
The concept of reflexivity naturally extends to locally convex spaces (cf. [[Locally convex space|Locally convex space]]).
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The concept of reflexivity naturally extends to [[locally convex space]]s.
 
 
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis" , Pergamon  (1982)  (Translated from Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis" , Pergamon  (1982)  (Translated from Russian)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Beauzamy,  "Introduction to Banach spaces and their geometry" , North-Holland  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.M. Day,  "Normed linear spaces" , Springer  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. van Dulst,  "Reflexive and superreflexive Banach spaces" , ''MC Tracts'' , '''102''' , Math. Centre  (1978)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Beauzamy,  "Introduction to Banach spaces and their geometry" , North-Holland  (1982)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M.M. Day,  "Normed linear spaces" , Springer  (1973)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  D. van Dulst,  "Reflexive and superreflexive Banach spaces" , ''MC Tracts'' , '''102''' , Math. Centre  (1978)</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 18:18, 26 September 2017

A Banach space $X$ that coincides under the canonical imbedding with its second dual $X^{{*}{*}}$ (cf. Adjoint space). More precisely, let $X^{*}$ be the space dual to $X$, i.e. the set of all continuous linear functionals defined on $X$. If $(x,f)$ is the value of the functional $f \in X^{*}$ on an element $x \in X$, then with a fixed $x$ and $f$ running through $X^{*}$, the formula $(x,f) = \mathcal{F}_x(f)$ defines a linear functional on $X^{*}$, that is, an element of the space $X^{{*}{*}}$. Let $\tilde X \subseteq X^{{*}{*}}$ be the set of such functionals. The correspondence $x \mapsto \mathcal{F}_x$ is an isomorphism which does not change the norm: $\Vert x \Vert = \Vert \mathcal{F}_x \Vert$. If $\tilde X = X^{{*}{*}}$, then the space $X$ is called reflexive. The spaces $\ell_p$ and $L_p(a,b)$, $p>1$, are reflexive, and the space $C[a,b]$ is not reflexive.

A space $X$ is reflexive if and only if the space $X^{*}$ is reflexive. Another criteria of reflexivity of a Banach space $X$ is weak compactness (cf. Weak topology) of the unit ball of this space.

A reflexive space is weakly complete and a closed subspace of a reflexive space is reflexive.

The concept of reflexivity naturally extends to locally convex spaces.

References

[1] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[2] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1
[3] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)


Comments

References

[a1] B. Beauzamy, "Introduction to Banach spaces and their geometry" , North-Holland (1982)
[a2] M.M. Day, "Normed linear spaces" , Springer (1973)
[a3] D. van Dulst, "Reflexive and superreflexive Banach spaces" , MC Tracts , 102 , Math. Centre (1978)
How to Cite This Entry:
Reflexive space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reflexive_space&oldid=41979
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article