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  2. Zorn's lemma - Wikipedia

    en.wikipedia.org/wiki/Zorn's_lemma

    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 HahnBanach ...

  3. Hahn–Banach theorem - Wikipedia

    en.wikipedia.org/wiki/HahnBanach_theorem

    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 HahnBanach theorem can be derived, was proved in ...

  4. Axiom of choice - Wikipedia

    en.wikipedia.org/wiki/Axiom_of_choice

    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.

  5. Axiom of dependent choice - Wikipedia

    en.wikipedia.org/wiki/Axiom_of_dependent_choice

    Furthermore, is equivalent to a weakened form of Zorn's lemma; specifically is equivalent to the statement that any partial order such that every well-ordered chain is finite and bounded, must have a maximal element. [3]

  6. Uniform boundedness principle - Wikipedia

    en.wikipedia.org/wiki/Uniform_boundedness_principle

    Together with the HahnBanach 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 .

  7. Functional analysis - Wikipedia

    en.wikipedia.org/wiki/Functional_analysis

    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 HahnBanach theorem, usually proved using the axiom of choice, although the strictly weaker Boolean prime ideal theorem ...

  8. Boolean prime ideal theorem - Wikipedia

    en.wikipedia.org/wiki/Boolean_prime_ideal_theorem

    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.

  9. List of theorems - Wikipedia

    en.wikipedia.org/wiki/List_of_theorems

    Hahn decomposition theorem (measure theory) Hahn embedding theorem (ordered groups) Hairy ball theorem (algebraic topology) HahnBanach theorem (functional analysis) Hahn–Kolmogorov theorem (measure theory) Hahn–Mazurkiewicz theorem (continuum theory) Hajnal–Szemerédi theorem (graph theory) Hales–Jewett theorem (combinatorics)