(2)1 For every set X and for all functions f , g such that X ⊆ dom f and f ⊆ g holds f X = g X . (3) For every non empty set A and for every set b such that A 6= {b} there exists an element a of A such that a 6= b. (4) For all sets X , Y holds every non empty subset of X→̇Y is a non empty functional set. (5) Let B be a non empty functional set and f be a function. Suppose f = ⋃ B. Then dom f = ⋃...

Few methods are known to construct nonLebesgue-measurable sets of reals: most standard ones start from a well-ordering of R, or from the existence of a non-trivial ultrafilter over ω, and thus need the axiom of choice AC or at least the Boolean Prime Ideal theorem (BPI see [5]). In this paper we present a new way for proving the existence of non-measurable sets using a convenient operation of a...

This paper will introduce and prove several theorems involving the separation of convex sets by hyperplanes, along with other interesting related results. It will begin with some basic separation results in Rn, such as the Hyperplane Separation Theorem of Hermann Minkowski, and then it will focus on and prove the extension of this theorem into normed vector spaces, known as the Hahn-Banach Sepa...

§0. Introduction. Few methods are known to construct non Lebesgue-measurable sets of reals: most standard ones start from a well-ordering of R, or from the existence of a non-trivial ultrafilter over ω, and thus need the axiom of choice AC or at least the Boolean Prime Ideal theorem BPI (see [5]). In this paper we present a new way for proving the existence of non-measurable sets using a conven...

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/notes 2012-13/05 banach.pdf] 1. Basic definitions 2. Riesz’ Lemma 3. Counter-example: non-existence of norm-minimizing element 4. Normed spaces of continuous linear maps 5. Dual spaces of normed spaces 6. Banach-Steinhaus/uniform-boundedness theorem 7. Open mapping theore...

Several generalizations of the Hahn–Banach extension theorem to K-convex multifunctions were stated recently in the literature. In this note we provide an easy direct proof for the multifunction version of the Hahn–Banach–Kantorovich theorem and show that in a quite general situation it can be obtained from existing results. Then we derive the Yang extension theorem using a similar proof as wel...

We describe an implementation of a pointfree proof of the Alaoglu and the Hahn-Banach theorems in Type Theory. The proofs described here are formalisations of the proofs presented in \The Hahn-Banach Theorem in Type Theory" 4]. The implementation was partially developed simultaneously with 4] and it was a help in the development of the informal proofs.

1. Basic definitions 2. Riesz’ Lemma 3. Counter-examples for unique norm-minimizing element 4. Normed spaces of continuous linear maps 5. Dual spaces of normed spaces 6. Baire’s theorem 7. Banach-Steinhaus/uniform-boundedness theorem 8. Open mapping theorem 9. Closed graph theorem 10. Hahn-Banach theorem Many natural spaces of functions, such as C(K) for K compact, and C[a, b], have natural str...

The idea behind this article is to provide a unified and relatively nontechnical framework for treating the main existence theorems for continuous linear functionals in functional analysis, convex analysis, Lagrange multiplier theory, and minimax theory. While many of the results in this article are already known, our approach is new, and gives a large number of results with considerable econom...