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Anywhere, not only in quantum mechanics, where one deals with operators, the space on which these operators operate must be specified; otherwise, one would be dealing with undefined objects. In quantum mechanics, Hilbert space (a complete inner-product space) plays a central role in view of the interpretation associated with wave functions: absolute value of each wave function is interpreted as being a probability distribution function. Probability distributions are by definition non-negative and normalizable (to unity). It is thus that Hilbert space takes a central position in quantum mechanics. I should add that in scattering theory (within the framework of quantum mechanics) one deals with functions that are not normalizable. Here, one deals with fluxes (that is, fluxes of incident and scattered particles), which must be well defined.
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