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.
Spectrum (functional analysis)
For example, consider the Hilbert space formula_24, that consists of all bi-infinite sequences of real numbers
that have a finite sum of squares formula_26. The bilateral shift operator formula_2 simply displaces every element of the sequence by one position; namely if formula_28 then formula_29 for every integer formula_30. The eigenvalue equation formula_31 has no solution in this space, since it implies that all the values formula_32 have the same absolute value (if formula_33) or are a geometric progression (if formula_34); either way, the sum of their squares would not be finite. However, the operator formula_20 is not invertible if formula_36. For example, the sequence formula_37 such that formula_38 is in formula_24; but there is no sequence formula_40 in formula_24 such that formula_42 (that is, formula_43 for all formula_30).