The Hilbert space

A Hilbert space X, as abstractly defined by von Neumann, is a linear strictly positive inner product space (generally over the field 3 of complex which is complete with respect to the metric generated by the -- inner product and which is separable. Its elements are called uectors, usually denoted by #, 9,. . . , and their inner or scalar product is denoted by (cp,#), whereas the elements of 9 are called scalars and usually denoted by a, b,. . . . In his work on linear integral equations (1904-1910) David Hilbert had studied two realizations of such a space, the Lebesgue space C2 of (classes of) all complex-valued Lebesgue measurable square-integrable functions on an interval of the real line R (or R itself), and the space l2 of sequences of complex numbers, the sum of whose absolute squares converges. Impressed by the fact that by virtue of the Riesz-Fischer theorem these two spaces can be shown to be isomorphic (and isometric) and hence, in spite of their apparent dissimilarity, to be essentially the same space, von Neumann named all spaces of this structure after Hilbert. The fact that this isomorphism entails the equivalence between Heisenberg's matrix mechanics and Schrodinner's wave - mechanics made von Neumann aware of the importance of Hilbert spaces for the mathematical formulation of quantum mechanics.

Reference: https://philosophy-of-sciences.blogspot.com/2018/12/the-philosophy-of-quantum-mechanics.html