Double dual space is isomorphic to vector space - Intuition

Let $x \in V$ and let $z \in V^*$ (where $V$ is a vector space). You can think of $z$ as doing something to $x$ (in other words, $z$ takes $x$ as input and returns $z(x)$ as output). But, you can equally well think of $x$ as doing something to $z$! In other words, you can imagine that $x$ itself takes $z$ as input and returns $z(x)$ as output. From this viewpoint, $x$ is a linear functional on $V^*$.

I like the notation \begin{equation} z(x) = \langle z, x \rangle, \end{equation} because it treats $z$ and $x$ symmetrically, and emphasizes that both viewpoints are equally valid.