Representing a Banach space as a function space

Solution 1:

The third question always has a positive answer. Let $E$ be a Banach space and let $B_{E^*$}$ be the unit ball of the dual space $E^*$ endowed with the weak-$\ast$ continuous topology. Then, any $x \in E$ has a representation as a continuous function $\hat{x} \in C(B_{E^*})$, given by $\hat{x}(\lambda) = \lambda(x)$. Recall that $B_X$ is compact by Banach-Alaouglu.

I do not known a definitive answer to the other questions (identify the closed subspaces of $L^p$). Using Rademacher techniques/Khintchine inequalities you can definitely embed $\ell^2$ inside $L^p$ for $p < \infty$.