Hilbert space
The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory. Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras. In the 1940s, Israel Gelfand, Mark Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory.
Weak convergence (Hilbert space)
The Hilbert space formula_14 is the space of the square-integrable functions on the interval formula_15 equipped with the inner product defined by
(see L space). The sequence of functions formula_17 defined by
converges weakly to the zero function in formula_14, as the integral
tends to zero for any square-integrable function formula_21 on formula_15 when formula_23 goes to infinity, which is by Riemann–Lebesgue lemma, i.e.
Although formula_25 has an increasing number of 0's in formula_26 as formula_23 goes to infinity, it is of course not equal to the zero function for any formula_23. Note that formula_25 does not converge to 0 in the formula_30 or formula_31 norms. This dissimilarity is one of the reasons why this type of convergence is considered to be "weak."