I agree that there's are lots of proofs of Pythagoras's theorem on Euclidean space. The comment I replied to said that it follows "directly" from the definition of the dot product. That's all I was disagreeing with. They had missed that they were using some or other property of the dot product that was actually proved from Pythagoras in the first place, or some other non-trivial fact about Euclidean geometry.
And I certainly don't mean to imply anything about the importance of abstract inner product spaces. In fact my masters thesis was about Hilbert spaces. And I find it pretty interesting that you can prove something like Pythagoras on the inner product form I mentioned at the end of my last comment.
Ah yes, you're right of course, and I haven't really thought about it in this way before - that either you're based on "real world" geometry, in which case things are a bit harder to prove but make sense, or you're more abstract, in which case you can define things to be easier to prove e.g. Pythagoras, but the complexity is in the mapping between your definitions and the "real world".
> In fact my masters thesis was about Hilbert spaces. And I find it pretty interesting that you can prove something like Pythagoras on the inner product form I mentioned at the end of my last comment.
That's pretty cool, you're definitely more knowledgeable than I am, I'm just a math amateur :)
And I certainly don't mean to imply anything about the importance of abstract inner product spaces. In fact my masters thesis was about Hilbert spaces. And I find it pretty interesting that you can prove something like Pythagoras on the inner product form I mentioned at the end of my last comment.