Looking for an easy lightning introduction to Hilbert spaces and Banach spaces
Solution 1:
I don't know how useful this will be, but I have some lecture notes that motivate the last three things on your list by first reinterpreting the finite dimensional spectral theorem in terms of the functional calculus. (There is also a section on the spectral theorem for compact operators, but this is just pulled from Zimmer's Essential Results of Functional Analysis.) I gave these lectures at the end of an undergraduate course on functional analysis, though, so they assume familiarity with Banach and Hilbert spaces.