Hilbert C*-module
Hilbert C*-modules are mathematical objects which generalise the notion of a Hilbert space (which itself is a generalisation of Euclidean space), in that they endow a linear space with an "inner product" which takes values in a C*-algebra. Hilbert C*-modules were first introduced in the work of Irving Kaplansky in 1953, which developed the theory for commutative, unital algebras (though Kaplansky observed that the assumption of a unit element was not "vital"). In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke and Marc Rieffel, the latter in a paper which used Hilbert C*-modules to construct a theory of induced representations of C*-algebras. Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, and provide the right framework to extend the notion of Morita equivalence to C*-algebras. They can be viewed as the generalization of vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C*-algebraic quantum group theory, and groupoid C*-algebras.
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."