# 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.

#### 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
How to Cite This Entry:
Second dual space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Second_dual_space&oldid=48640
This article was adapted from an original article by M.I. Kadets (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article