This document presents a structured exposition of the mathematical framework underlying quantum mechanics. The text revisits the emergence of Hilbert spaces as the natural setting for quantum theory and discusses the development of the subject through the ideas of Hilbert, Fréchet, and von Neumann. Particular attention is given to the sequence space $\ell^2$, orthogonal decompositions, completeness, and the role of self-adjoint operators and spectral theory in the mathematical formulation of quantum mechanics. The aim of the document is to provide a clear and rigorous reference text for students approaching the mathematical structures of quantum theory. The presentation emphasizes conceptual clarity while maintaining the formal precision required in functional analysis and mathematical physics.