A reflexive space is a normed space 
for which the canonical map 
into its second dual
normed space 
is a surjection. Here, 
is the continuous dual
normed space of 
,
and 
for 
.
This canonical map is an embedding
and a linear transformation that is an isometry. This concept was introduced by Hahn (1927).
For example, finite-dimensional normed spaces and Hilbert
spaces are reflexive. The space 
of absolutely summable sequences
of complex numbers is not reflexive. James (1951)
constructed a non-reflexive Banach space that is
isometrically isomorphic
to its second conjugate space.
Reflexive spaces are Banach spaces. This follows since given a normed space 
that may or may not be Banach,
the norm on 
induces a norm (called the dual norm) on the dual 
of 
, and under the dual norm, 
is Banach. Iterating again,
(the bidual of 
) is also Banach, and since
is reflexive if it coincides with its
bidual, 
is Banach.
