Functional analysis
Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis. Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to formula_1. Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have already been proven.
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."