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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 asserts that φ can be extended to a linear functional on V that is dominated by N. To derive this from the M. Riesz extension theorem, define a convex cone K ⊂ R×V by = {(,) ()}. Define a functional φ 1 on R×U by
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.