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.
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.
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.