• Pre-Hilbert spaces: definition • Cauchy-Schwarz-Bunyakowski inequality • Example: spaces ` • Triangle inequality, associated metric, continuity issues • Hilbert spaces, completions, infinite sums • Minimum principle • Orthogonal projections to closed subspaces • Orthogonal complements W⊥ • Riesz-Fischer theorem on linear functionals • Orthonormal sets, separability • Parseval equality, Bessel...

An interesting result proved by Halmos in Hal (Michigan Mathematical Journal, 15, 215–223 (1968) is that the set of irreducible operators dense $${\mathcal {B}}({\mathcal {H}})$$ sense Hilbert-Schmidt approximation. In a von Neumann algebra {M}}$$ with separable predual, an operator $$a\in {\mathcal said to be if $$W^*(a)$$ subfactor , i.e., $$W^*(a)'\cap {M}}={{\mathbb {C}}} \cdot I$$ . Let $$...