# 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 |

**How to Cite This Entry:**

Second dual space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Second_dual_space&oldid=48640