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 →

## Mathematical exceptions to the rules or intuition