Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. It's essentially just an infinitely dimensional vector space. ﻿A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.
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."