# 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.
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.