Suppose $V$ is a real vector space.
Then $V^*$, its dual space, is the vector space of linear maps $V\to \mathbb R$
How then do I interpret $(V^*)^*$, the dual space of the dual space?
Suppose $V$ is a real vector space.
Then $V^*$, its dual space, is the vector space of linear maps $V\to \mathbb R$
How then do I interpret $(V^*)^*$, the dual space of the dual space?
The space of linear maps $\ell : V \rightarrow \mathbb{R}$ is itself a vector space, with pointwise addition and scalar multiplication of functions. Thus, $(V^*)^*$ is the dual of this vector space.
There is a canonical linear transformation $\xi : V \rightarrow (V^*)^*$ defined by $\xi(v) = \xi_v$, where $\xi_v : V^* \rightarrow \mathbb{R}$ is the linear map given by $\xi_v(\ell) = \ell(v)$. The map $\xi$ is injective, so when $V$ is a finite dimensional vector space, the map $\xi$ is a (canonical) isomorphism $V \cong (V^*)^*$. However, $\xi$ is not necessarily an isomorphism if $V$ is infinite dimensional.