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The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s.The special case of the theorem for the space [,] of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, [1] and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in ...
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
If Y is an injective Banach space, then for every Banach space X, every continuous linear operator from a vector subspace of X into Y has a continuous linear extension to all of X. [1] In 1953, Alexander Grothendieck showed that any Banach space with the extension property is either finite-dimensional or else not separable. [1]
Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.
Hahn–Banach dominated extension theorem [18] (Rudin 1991, Th. 3.2) — If : is a sublinear function, and : is a linear functional on a linear subspace which is dominated by p on M, then there exists a linear extension : of f to the whole space X that is dominated by p, i.e., there exists a linear functional F such that = for all , and | | for ...
Theorem [13] (Kalton) — Every complete metrizable TVS with the Hahn-Banach extension property is locally convex. If a vector space X {\displaystyle X} has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.
The Hahn–Banach theorem in functional analysis, allowing the extension of linear functionals. The theorem that every Hilbert space has an orthonormal basis. The Banach–Alaoglu theorem about compactness of sets of functionals.
By the Hahn–Banach theorem the latter admits a norm-one linear extension on ℓ ∞ (Z), which is by construction a shift-invariant finitely additive probability measure on Z. If every conjugacy class in a locally compact group has compact closure, then the group is amenable.