Reflexive Space -- from Wolfram MathWorld
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Created, developed, and nurtured by Eric Weisstein at Wolfram Research
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Reflexive Space
Let
For example, finite-dimensional (normed) spaces and Hilbert spaces are reflexive. The space
Reflexive spaces are Banach spaces. This follows since given a normed space
SEE ALSO: Banach Space, Dual
Vector Space, Normed Space
Portions of this entry contributed by Mohammad Sal Moslehian Portions of this entry contributed by Christopher Stover Hahn, H. "Über lineare Gleichungssysteme in linearen Räumen." J. reine angew. Math. 157, 214-229, 1927. James, R. C. "A Non-Reflexive Banach Space Isometric with Its Second Conjugate Space." Proc. Nat. Acad. Sci. USA 37, 174-177, 1951. Moslehian, Mohammad Sal; Stover, Christopher; and Weisstein, Eric W. "Reflexive Space." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ReflexiveSpace.html Wolfram Web Resources
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be a 
denote the second 
defined by 
gives an isometric linear isomorphism (embedding) from 
. The space
of absolutely
summable complex sequences is not reflexive. James (1951) constructed a non-reflexive
Banach space that is isometrically isomorphic to its second conjugate space.
of 
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{1/4, -1/2, 1} cross {1/3, 1, -2/3}