This is perhaps an inferior example to those already mentioned, but my understanding of the so-called Grothendieck double-limit criterion (used to test if a map from a Banach space $X$ to a dual Banach space $Y^*$ is weakly compact) is that the proof proceeds along the following lines:
if $X$ and $Y$ are Banach spaces we view them as closed subspaces of $C(\Omega_X)$ and $C(\Omega_Y)$ where $\Omega_X$ is the closed unit ball of $X^*$ equipped with the weak-star topology, and likewise for $\Omega_Y$;
taking a bilinear map $X\times Y \to {\bf C}$, we appeal to Hahn-Banach to lift/extend it to a bilinear map $C(\Omega_X)\times C(\Omega_Y)\to {\bf C}$
now we are in a position to exploit things like Riesz representation, positivity, and ideas/tools from measure theory.
As will be seen from the sketchiness of this account, I may have misunderstood or misremembered what is going on. But the gist of my claim is that a non-trivial theorem about Banach spaces is proved by embedding a given Banach space into one which seems nicer, namely $C(\Omega)$.
