Second dual space

The space $ X ^ {\prime\prime} $ dual to the space $ X ^ \prime $, where $ X ^ \prime $ is the strong dual to a Hausdorff locally convex space $ X $, i.e. $ X ^ \prime $ is equipped with the strong topology. Each element $ x \in X $ generates an element $ F \in X ^ {\prime\prime} $ in accordance with the formula $ F( f ) = f( x) $( $ f \in X ^ \prime $). If $ X ^ {\prime\prime} = X $, the space $ X $ is semi-reflexive. If $ X $ is a barrelled space, the linear mapping $ \pi : X \rightarrow X ^ {\prime\prime} $ defined by $ \pi ( x)= F $ is an isomorphic imbedding of the space $ X $ into the space $ X ^ {\prime\prime} $. The imbedding $ \pi $ is called canonical. For normed spaces $ \pi $ is an isometric imbedding.

Comments

The second dual $ X ^ {\prime\prime} = ( X ^ \prime ) ^ \prime $ is also called the bidual.

For (semi-) reflexivity see also Reflexive space. For the (first) dual space see Adjoint space. The space $ X $ is reflexive if the canonical imbedding $ X \rightarrow X ^ {\prime\prime} $ is surjective and also the two topologies coincide, where $ X ^ {\prime\prime} $ is given the strong topology defined by the dual pair $ ( X ^ \prime , X ^ {\prime\prime} ) $. For Banach spaces semi-reflexivity is the same as reflexivity.

References

[a1]	D. van Dulst, "Reflexive and superreflexive spaces" , MC Tracts , 102 , Math. Centre (1978)
[a2]	G. Köthe, "Topological vector spaces" , 1 , Springer (1969) pp. §23.5