Let 
be a normed space and 
denote the second dual
vector space of 
.
The canonical map 
defined by 
gives an isometric linear isomorphism (embedding)
from 
into 
.
The space 
is called reflexive if this map is surjective. This concept was introduced by Hahn
(1927).
For example, finite-dimensional (normed) spaces and Hilbert spaces are reflexive. 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.
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.
