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Attempts to find classes of locally convex topological vector spaces on which the uniform boundedness principle holds eventually led to barrelled spaces. That is, the least restrictive setting for the uniform boundedness principle is a barrelled space, where the following generalized version of the theorem holds (Bourbaki 1987, Theorem III.2.1):
In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are always valued in the real numbers or the complex numbers) to linear operators valued in topological vector spaces (TVSs).
Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs; Continuous linear operator; Densely defined operator – Function that is defined almost everywhere (mathematics) Hahn–Banach theorem – Theorem on extension of bounded linear functionals
Theorem — Let X be a Banach space, C be a compact operator acting on X, and σ(C) be the spectrum of C. Every nonzero λ ∈ σ ( C ) is an eigenvalue of C . For all nonzero λ ∈ σ ( C ), there exist m such that Ker (( λ − C ) m ) = Ker (( λ − C ) m +1 ), and this subspace is finite-dimensional.
The Banach–Stone theorem has some generalizations for vector-valued continuous functions on compact, Hausdorff topological spaces. For example, if E is a Banach space with trivial centralizer and X and Y are compact, then every linear isometry of C(X; E) onto C(Y; E) is a strong Banach–Stone map.
If A is a group algebra for some locally compact group G then A is amenable if and only if G is amenable.; If A is a C*-algebra then A is amenable if and only if it is nuclear.; If A is a uniform algebra on a compact Hausdorff space then A is amenable if and only if it is trivial (i.e. the algebra C(X) of all continuous complex functions on X).
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm.
The spectral mapping theorem states: if x is an element of a unital Banach algebra and f is a holomorphic function in a neighborhood of the spectrum () of x, then (()) = (()), where () is an element of the Banach algebra defined via the Cauchy's integral formula.