Let T̃ = ∑n i=1 ũi⊗ vi : V → V = [v1, ..., vn] ⊂ X, where ũi ∈ V ∗ and X is a Banach space. Let T = ∑n i=1 ui ⊗ vi : X → V be an extension of T̃ to all of X (i.e., ui ∈ X∗) such that T has minimal (operator) norm. In this paper we show in particular that, in the case n = 2 and the field is R, there exists a rank-n T̃ such that ‖T‖ = ‖T̃‖ for all X if and only if the unit ball of V is either not smo...

§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...

was established by Hahn [1; p. 217] in 1927, and independently by Banach [2; p. 212] in 1929, who also generalized Theorem 0 for real spaces, to the situation in which the functional q :E^>R is an arbitrary subadditive, positive homogeneous functional [2; p. 226]. Theorem 0 was not established for complex spaces until 1938, when it was deduced from the real theorem by Bohnenblust and Sobczyk [3...

In the present paper, we study some properties of fuzzy norm of linear operators. At first the bounded inverse theorem on fuzzy normed linear spaces is investigated. Then, we prove Hahn Banach theorem, uniform boundedness theorem and closed graph theorem on fuzzy normed linear spaces. Finally the set of all compact operators on these spaces is studied.

Borwein’s norm duality theorem establishes the equality between the outer (inner) norm of a sublinear mapping and the inner (outer) norm of its adjoint mappings. In this note we provide an extended version of this theorem with a new and self-contained proof relying only on the Hahn-Banach theorem. We also give examples showing that the assumptions of the theorem cannot be relaxed.