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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 ...
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.
The existence of a shift-invariant, finitely additive probability measure on the group Z also follows easily from the Hahn–Banach theorem this way. Let S be the shift operator on the sequence space ℓ ∞ ( Z ), which is defined by ( Sx ) i = x i +1 for all x ∈ ℓ ∞ ( Z ), and let u ∈ ℓ ∞ ( Z ) be the constant sequence u i = 1 for ...
The above theorem can be used to extend a bounded linear transformation : to a bounded linear transformation from ¯ = to , if is dense in . If S {\displaystyle S} is not dense in X , {\displaystyle X,} then the Hahn–Banach theorem may sometimes be used to show that an extension exists .
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.
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 ...
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.