(2) is fine, but since V is a finite dimensional inner product space, it makes no difference if it is only F^{dimension}.

Spectral theorems are far too important for noone to try to generalize There is a version about self-adjoint operators in infinite dimensional Hilbert spaces (which is a direct generalization of the finite-d case, and is used very finely in proving existence of solutions for the Laplacian operator) and a very unexpected version about "completely continuous" operators in the same setting, which leads to the so-called Functional Calculus.

Alright, the thing is Spectral Theorems are still being worked on!