The second theorem (the generalized Pythagorean theorem, which uses inner product spaces) feels incomplete to me. The exposition is fine, but I don't think the explanation should have concluded with the regular
a^2 + b^2 = c^2
equality. You wouldn't use that in the context of an inner product space. The generalized Pythagorean theorem actually looks like
||x + y||^2 = ||x||^2 + ||y||^2 + 2<x, y>.
I think it would be better if there's less editorializing of the equalities and more relating of their abstract forms to their "simple", familiar forms. Otherwise I don't really understand the exercise - if the first page has a theorem involving inner product spaces, this is clearly targeted at people with at least a full course of linear algebra under their belt. Unless the audience is already aware of how that second equality implies the first equality, this explanation doesn't capture the heart of it.
As another commenter said, this is not a self-contained exposition (and I don't think it realistically could be). But if it's not going to be self-contained, I think it could be improved by more completely showing how the elegant abstractions imply the things we're already familiar with in a neat way.
The Pythagorean theorem is about orthogonal vectors, so it would just be
|x + y|^2 = |x|^2 + |y|^2
Plus the line just above that tells you: let
a = |x|, b = |y|, c = |x-y|
I don't think they are obscuring anything. I think they are showing how the a b and c in the familiar a^2+b^2=c^2 can be generalised to |x|, |y|, |x-y| for any orthogonal vectors x and y in an inner product space.
As another commenter said, this is not a self-contained exposition (and I don't think it realistically could be). But if it's not going to be self-contained, I think it could be improved by more completely showing how the elegant abstractions imply the things we're already familiar with in a neat way.