# A vector space not isomorphic to its double dual

In this page $$\mathbb{F}$$ refers to a field. Given any vector space $$V$$ over $$\mathbb{F}$$, the dual space $$V^*$$ is defined as the set of all linear functionals $$f: V \mapsto \mathbb{F}$$. The dual space $$V^*$$ itself becomes a vector space over $$\mathbb{F}$$ when equipped with the following addition and scalar multiplication:
$\left\{ \begin{array}{lll}(\varphi + \psi)(x) & = & \varphi(x) + \psi(x) \\ (a \varphi)(x) & = & a (\varphi(x)) \end{array} \right.$ for all $$\phi, \psi \in V^*$$, $$x \in V$$, and $$a \in \mathbb{F}$$.
There is a natural homomorphism $$\Phi$$ from $$V$$ into the double dual $$V^{**}$$, defined by $$(\Phi(v))(\phi) = \phi(v)$$ for all $$v \in V$$, $$\phi \in V^*$$. This map $$\Phi$$ is always injective. Continue reading A vector space not isomorphic to its double dual