It's well known that if $V$ is a vector space over a field $F$, then there is a natural injection from $V$ to the double dual $V^{**}$, which associates to every $v \in V$ the evaluation map $\phi \mapsto \phi(v)$, where $\phi: V \to F$ is an arbitrary functional in $V^*$. It's also well known that this injection is an isomorphism if $V$ is finite-dimensional, as any finite-dimensional vector space has the same dimension as its dual.
My question is this: are there any nice, readily understood examples of infinite-dimensional vector spaces $V$ for which an element of $V^{**}$ that is not an evaluation map can be explicitly constructed (at least with the axiom of choice)? I find infinite-dimensional double dual spaces hard even to think about.
Quick example, using the axiom of choice: take $V$ to be the set of polynomials with real coefficients. Let $S$ denote the subspace of $V^*$ consisting of those functionals such that $\lim_{n \to \infty} f(x^n)$ exists. With the axiom of choice, there necessarily exists a complementary subspace $S'$ such that $V = S \oplus S'$.
Define $\phi:S \to \Bbb R$ by $\phi(f) = \lim_{n \to \infty} f(x^n)$. Define $\phi:S' \to \Bbb R$ by $\phi(f) = 0$. Extend $\phi$ to all of $V$ by linearity.