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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 ...
Zorn's lemma is also equivalent to the strong completeness theorem of first-order logic. [23] Moreover, Zorn's lemma (or one of its equivalent forms) implies some major results in other mathematical areas. For example, Banach's extension theorem which is used to prove one of the most fundamental results in functional analysis, the Hahn–Banach ...
Zorn's lemma, the axiom of choice, and Tychonoff's theorem can all be used to prove the ultrafilter lemma. The ultrafilter lemma is strictly weaker than the axiom of choice. The ultrafilter lemma has many applications in topology. The ultrafilter lemma can be used to prove the Hahn-Banach theorem and the Alexander subbase theorem.
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points.
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.
Any Banach limit on is an example of an element of the dual Banach space of which is not in . The dual of ℓ ∞ {\displaystyle \ell ^{\infty }} is known as the ba space , and consists of all ( signed ) finitely additive measures on the sigma-algebra of all subsets of the natural numbers , or equivalently, all (signed) Borel measures on the ...
Theorem IV.7.7; The separable Hahn–Banach theorem in the form: a bounded linear form on a subspace of a separable Banach space extends to a bounded linear form on the whole space. The Jordan curve theorem; Gödel's completeness theorem (for a countable language). Determinacy for open (or even clopen) games on {0,1} of length ω.
Hahn–Banach The Hahn–Banach theorem states: given a linear functional ℓ {\displaystyle \ell } on a subspace of a complex vector space V , if the absolute value of ℓ {\displaystyle \ell } is bounded above by a seminorm on V , then it extends to a linear functional on V still bounded by the seminorm.