The Hahn-Banach theorem is one of the most fundamental theorems in the functional analysis theory. This theorem is well known in the case where the range space is the real number system as follows. Let p be a sublinear mapping from a vector space X into the real number system R, Y a subspace of X, and q a linear mapping from Y into R such that q ≤ p on Y. Then there exists a linear mapping g fr...

We continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach...

We give the basic de nitions for pointfree functional analysis and present constructive proofs of the Alaoglu and Hahn Banach theorems in the set ting of formal topology Introduction We present the basic concepts and de nitions needed in a pointfree approach to functional analysis via formal topology Our main results are the constructive proofs of localic formulations of the Alaoglu and Helly H...

We establish, in innnite dimensional Banach space, a nonconvex separation property for general closed sets that is an extension of Hahn-Banach separation theorem. We provide some consequences in optimization, in particular the existence of singular multipliers and show the relation of our principle with the extremal principle of Mordukhovich.

Fuzzy anti-bounded linear functional and fuzzy antidual spaces are defined. Hahn-Banach theorem and some of its consequences on fuzzy anti-normed linear space are studied. Two fundamental theorems; namely, open mapping theorem and closed graph theorem are established. Keywords-Fuzzy anti-norm, α-norm, Fuzzy anti-complete, Fuzzy anti-bounded linear functional, Fuzzy anti-dual space.

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...

Some new results which generalize the Hahn-Banach theorem from scalar or vector-valued case to set-valued case are obtained. The existence of the Borwein-strong subgradient and Yang-weak subgradient for set-valued maps are also proven. we present a new Lagrange multiplier theorem and a new Sandwich theorem for set-valued maps.