The dual of a vector space is space of linear functions from the vector space to the underlying field. For vector spaces of finite dimension they are isomorphic, with the dot product (curried) as the transformation from one to the other.

\[ v \in V \mapsto (\_ \cdot v) \in V* \]

I don't remember what happens for infinite dimensional spaces.

\[ v \in V \mapsto (\_ \cdot v) \in V* \]

I don't remember what happens for infinite dimensional spaces.