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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 ...
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 ...
A standard application is the proof of the Picard–Lindelöf theorem about the existence and uniqueness of solutions to certain ordinary differential equations. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator on the space of continuous functions under the uniform norm. The Banach ...
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 .
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.
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.