The Schrödinger equation is a linear equation: linear superpositions of solutions are again solutions. We also need square integrability of the solutions. Thus one is naturally led to a vector space of square integrable functions for the space of wave functions. It turns out that this space has additional mathematical structure, namely the structure provided by an inner product.
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Dürr, D., Teufel, S. (2009). Hilbert Space. In: Bohmian Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/b99978_13
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DOI: https://doi.org/10.1007/b99978_13
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