[MITSardine]

Sorry, I couldn't find the page in English, but what you're talking about is a Hilbert basis: https://fr.wikipedia.org/wiki/Base_de_Hilbert . There is a paragraph on this in the orthonormal basis English page: https://en.wikipedia.org/wiki/Orthonormal_basis

Another example is the eigenvectors of linear operators like the Laplacian. Recall how, in finite dimension, the eigenvectors of a full rank operator (matrix) form an orthonormal basis of the vector space. There is a similar notion in infinite dimension. I can't find an English page that covers this very well, but there's a couple of paragraphs in the Spectral Theorem page (https://en.wikipedia.org/wiki/Spectral_theorem#Unbounded_sel... ). The article linked here also touches on this.

Regarding your last sentence, one thing to note is that having a basis is not what makes you a Hilbert space, but rather having an inner product! In fact, to get the Fourier coefficients, you need to use that inner product.

[seanhunter]

That's awesome info thank you so much. Reading it, a Hilbert basis is exactly what I am talking about. It's always exciting when my intuition guides me on the right path. I'll check out the Spectral theorem page also.