A new & peculiar essay, on a (very!) unusual approach I've been using to deepen my understanding of mathematics: http://cognitivemedium.com/srs-mathematics
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That lets you deduce the spectral theorem for normal operators from the spectral theorem for self-adjoint operators and simultaneous diagonalization for commuting operators. That's been my preferred proof for a while, but everyone has their pet favorites for these basic theorems.
Amusingly, that's one of the proofs I discovered (and forgot) on my walk. The proof of diagonalizability for self-adjoint matrices is essentially obvious. The commuting => simultaneous basis needs a little more care, though.
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(I did _not_ think at all about operators, though.)
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Inductive proofs, like that in my essay, are harder to make work in infinite dimensions!
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