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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.
The most important among them are Zorn's lemma and the well-ordering theorem. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem. Set theory. Tarski's theorem about choice: For every infinite set A, there is a bijective map between the sets A and A×A.
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 Banach limits is usually proved using the Hahn–Banach theorem (analyst's approach), [1] or using ultrafilters (this approach is more frequent in set-theoretical expositions). [2] These proofs necessarily use the axiom of choice (so called non-effective proof).
Goldstine theorem (functional analysis) Hahn–Banach theorem (functional analysis) Hilbert projection theorem (convex analysis) Kachurovskii's theorem (convex analysis) Kirszbraun theorem (Lipschitz continuity) M. Riesz extension theorem (functional analysis) Milman–Pettis theorem (Banach space) Moore–Aronszajn theorem (Hilbert space)
In its preface, Phelps advised readers of the prerequisite "background in functional analysis": "the main rule is the separation theorem (a.k.a. [also known as] the Hahn–Banach theorem): Like the standard advice given in mountaineering classes (concerning the all-important bowline for tying oneself into the end of the climbing rope), you ...