Reddit reviews First Course in Wavelets with Fourier Analysis
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It won't always be referred to as "Fourier's theorem". Some texts may refer to it as "completeness of fourier series", or they may prove more general versions of the theorem using Sturm–Liouville theory. Note that "completeness" has a technical definition, but in this context roughly means that any square-integrable periodic function can be described as a (possibly infinite) trigonometric series.
I seem to recall that A First Course in Wavelets with Fourier Analysis was readable & yet not long-winded, and had a fairly rigorous proof of Fourier's theorem by the middle of the second chapter. See Amazon.com. A free PDF from an MIT OpenCourseware that might be of use is here. I skimmed over it; it does prove what you are interested in, but I can't vouch for its readability.