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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I'm honestly not sure it will work for other people at all! But it's had a big impact on me, on those times I've tried it.
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It's based on a whole tonne of heuristics (many described in the essay) for really pulling the proof of a theorem apart, atomizing it into a million simple observations, a sort of proof network.
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The end result (ideally) is a state of what I think of as "being inside a piece of mathematics", a sense of being able to work directly with the objects & change my understanding of them in a way that doesn't directly use conventional symbolic or verbal or visual representationspic.twitter.com/Jmyo3bi6hA
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(Accidentally uploaded without a spellcheck, now fixed!)
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End of conversation
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Lakatos' "Proofs and Refutations" may be of use to you as you refine your views on these matters. Oh, and by the way: a normal linear operator is one whose (self-adjoint) real and imaginary parts commute. This points the way to a rather different proof from the one you describe.
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"If you take an idea that seems outstandingly creative, it usually didn’t appear in someone’s head by magic. Some thought process led to it, and this can be broken into smaller ideas until you get down to the lowest level of [the] thought process, which is essentially routine."
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