which consists of all continuous linear functionals and is equipped with the strong topology , that is, the topology of uniform convergence on bounded subsets in . The space is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space , which is called the strong bidual space for . It consists of all continuous linear functionals and is equipped with the strong topology . Each vector generates a map by the following formula: This is a continuous linear functional on , that is, . One obtains a map called the evaluation map or the canonical injection: which is a linear map. If is locally convex, from the Hahn–Banach theorem it follows that is injective and open (that is, for each neighbourhood of zero in there is a neighbourhood of zero in such that ). But it can be non-surjective and/or discontinuous. A locally convex space is called semi-reflexive if the evaluation map is surjective (hence bijective); it is called reflexive if the evaluation map is surjective and continuous, in which case will be an isomorphism of TVSs). Characterizations of semi-reflexive spaces If is a Hausdorff locally convex space then the following are equivalent: is semireflexive; the weak topology on had the Heine-Borel property (that is, for the weak topology , every closed and bounded subset of is weakly compact). If linear form on that continuous when has the strong dual topology, then it is continuous when has the weak topology; is barrelled, where the indicates the Mackey topology on ; weak the weak topology is quasi-complete. Sufficient conditions Every semi-Montel space is semi-reflexive and every Montel space is reflexive. Properties If is a Hausdorff locally convex space then the canonical injection from into its bidual is a topological embedding if and only if is infrabarrelled. The strong dual of a semireflexive space is barrelled. Every semi-reflexive space is quasi-complete. Every semi-reflexive normed space is a reflexive Banach space. The strong dual of a semireflexive space is barrelled. Reflexive spaces If is a Hausdorff locally convex space then the following are equivalent: is reflexive; is semireflexive and barrelled; is barrelled and the weak topology on had the Heine-Borel property (which means that for the weak topology , every closed and bounded subset of is weakly compact). is semireflexive and quasibarrelled. If is a normed space then the following are equivalent: