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Throughout my education I've encountered the following examples of Hilbert spaces: 1) the wave functions of a quantum mechanical system are elements of a Hilbert space 2) in the finite element method for solving PDEs solution can be approximated with functions which are elements of a Hilbert space.

Can you give me other examples where Hilbert spaces are encountered in Math/Science in general?

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Have you looked at the Wikipedia article (en.wikipedia.org/wiki/Hilbert_space)? It contains lots of examples. –  Fredrik Meyer Dec 8 '11 at 20:21

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The space of weight $k$ cusp forms is a Hilbert space; the inner product is the Petersson inner product.

Added: In view of your first example, I decided to add this comment. According to the introduction of the book Modular forms and string duality, it seems that modular forms (of which cusp forms are an important example) are related to different dualities and symmetries appearing in string theory, mainly via the arithmetic of Calabi-Yau manifolds/varieties.

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Reproducing Kernel Hilbert spaces are fundamental to many machine learning algorithms.

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