Necessity of Galerkin Approximation in Existence Proof for Parabolic PDEs I'm studying the proof of the existence of solutions to linear parabolic PDEs, e.g. the heat equation $u_t = \Delta u + f$. The standard technique seems to be to do a Galerkin approximation, see e.g. Section 6.5 in these notes. 
My question is: In the notes mentioned above, a proof of the existence of solutions in the finite dimensional case is given (see Proposition 6.5), but I do not see where this proof makes use of the fact that the system is finite dimensional. If this were not required, however, then what's the point in making a Galerkin approximation in the first place?

I think I've found the answer. The proof is based on showing that the map
$$
\Phi(u) := u_0 + \int_0^t \Delta u + f \, dt
$$
is a contraction and hence has a fixed point. In the infinite-dimensional problem, this definition of $\Phi$ doesn't make sense since $\Delta : H^1_0 \to H^{-1}$ and thus it is not clear what the domain and range of $\Phi$ should be. In the finite-dimensional system, on the other hand, we replace $\Delta$ by an operator $\Delta_n : \mathbb{R}^n \to \mathbb{R}^n$ and everything works out. 
