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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 ...
The proof is similar to the proof of the Hahn–Banach theorem (see also below). By transfinite induction or Zorn's lemma it is sufficient to consider the case dim E / F = 1 {\displaystyle E/F=1} .
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.
To show the existence of a vector space basis for such spaces may require Zorn's lemma. However, a somewhat different concept, the Schauder basis, is usually more relevant in functional analysis. Many theorems require the Hahn–Banach theorem, usually proved using the axiom of choice, although the strictly weaker Boolean prime ideal theorem ...
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.
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 Hahn–Banach theorem. [27] In ZF, the Hahn–Banach theorem is strictly weaker than the ultrafilter lemma. The Banach–Tarski paradox. In fact, under ZF, the Banach–Tarski paradox can be deduced from the Hahn–Banach theorem, [29] [30] which is strictly weaker than the Ultrafilter Lemma. Every set can be linearly ordered.