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.
Hilbert's fourteenth problem
gave the following counterexample to Hilbert's problem. The field "k" is a field containing 48 elements "a", ...,"a", for "i"=1, 2, 3 that are algebraically independent over the prime field. The ring "R" is the polynomial ring "k"["x"...,"x", "t"...,"t"] in 32 variables. The vector space "V" is a 13-dimensional vector space over "k" consisting of all vectors ("b"...,"b") in "k" orthogonal to each of the three vectors ("a", ...,"a") for "i"=1, 2, 3. The vector space "V" is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on "R" by fixing all elements "t" and taking "x" to "x" + "b""t". Then the ring of elements of "R" invariant under the action of the group "V" is not a finitely generated "k"-algebra.