View source for Reflexive space

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.

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.

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]]).

====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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====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>