Reflexive space
A Banach space that coincides under the canonical imbedding with its second dual
(cf. Adjoint space). More precisely, let
be the space dual to
, i.e. the set of all continuous linear functionals defined on
. If
is the value of the functional
on an element
, then with a fixed
and
running through
, the formula
defines a linear functional on
, that is, an element of the space
. Let
be the set of such functionals. The correspondence
is an isomorphism which does not change the norm:
. If
, then the space
is called reflexive. The spaces
and
,
, are reflexive, and the space
is not reflexive.
A space is reflexive if and only if the space
is reflexive. Another criteria of reflexivity of a Banach space
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 (cf. Locally convex space).
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) |
Reflexive space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reflexive_space&oldid=17039