Summary of contents: Hilbert spaces; norm induced by an inner product; proof of the Cauchy-Schwarz inequality; proof that the induced norm is a norm; detailed proof, step by step, of the Jordan - von Neumann theorem: a norm is induced by an inner product if and only if it satisfies the parallelogram identity, and the inner product is determined by the polarisation identity, adapting hints from Friedberg, Insel, Spence's Linear algebra; proof that the dual of a Hilbert space is a Hilbert space; proof that inner products are sequentially continuous; Hamel basis and dimension of a vector space; Schauder basis and properties; orthonormal Schauder basis and separable Hilbert spaces; coefficients with respect to a Schauder basis; Pythagoras' theorem; structure-preserving maps and classification of structures; unitary map (or unitary operator); proof that a surjective map which preserves the inner product is a unitary map; square-summable complex sequences; classification of separable Hilbert spaces: proof that every separable Hilbert space is unitarily equivalent to l^2(N).
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