In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from into its bidual (which is the strong dual of the strong dual of ) is a homeomorphism (or equivalently, a TVS isomorphism). A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this (always linear) evaluation map is an isometric isomorphism and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective are called semi-reflexive spaces.
In 1951, R. C. James discovered a Banach space, now known as James' space, that is not reflexive (meaning that the canonical evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such isometric isomorphism is necessarily not the canonical evaluation map). So importantly, for a Banach space to be reflexive, it is not enough for it to be isometrically isomorphic to its bidual; it is the canonical evaluation map in particular that has to be a homeomorphism.
Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. Reflexive Banach spaces are often characterized by their geometric properties.