In this note, we aim to present some properties of the space of all weakly fuzzy bounded linear operators, with the Bag and Samanta’s operator norm on Felbin’s-type fuzzy normed spaces. In particular, the completeness of this space is studied. By some counterexamples, it is shown that the inverse mapping theorem and the Banach-Steinhaus’s theorem, are not valid for this fuzzy setting. Also...

Given f : X → R ∪ {+∞} a convex and lower semi-continuous function defined on a reflexive Banach space X, and L, a closed linear manifold of X over which f takes at least a real value, the aim of this note is to prove the following Baire category result: in the Euclidean setting, the set of affine functions dominated by f on L for which there is no dominated extension to X is always of first Ba...

Recently, Huang and Ng presented second-order sufficient conditions for error bounds of continuous and Gâteaux differentiable functions in Banach spaces. Wu and Ye dropped the assumption of Huang and Ng on Gâteaux differentiability but required the space to be a Hilbert space. We carry on this research in two directions. First we extend Wu and Ye’s result to some non-Hilbert spaces; second, sam...

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $left(varphi_1, varphi_2right)$-convex function $g, $ with arbitrarily small norm,  such that $f + g $ attains its strong minimum on $X. $ This result extends some of the  well-known varitional principles as that of Ekeland [On the variational principle,  J. Ma...

Denote by [0, ω1) the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let C0[0, ω1) be the Banach space of scalar-valued, continuous functions which are defined on [0, ω1) and vanish eventually. We show that a weak ∗compact subset of the dual space of C0[0, ω1) is either uniformly Eberlein compact, or it contains a homeomorphic copy of...

Let (Ω,Σ) be a measurable space and let X be a Banach space. Throughout this paper G will be a countably additive vector measure G : Σ → X. Consider the space L1(G) of (classes of real) G-integrable functions, following the definition of Bartle, Dunford and Schwartz [1] and Lewis [9]. The properties of this space have been studied by Kluvánek and Knowles [7], Okada [10] and Curbera [2]. In the ...

It is shown, using the Borwein–Preiss variational principle that for every continuous convex function f on a weakly compactly generated space X, every x0 ∈ X and every weakly compact convex symmetric set K such that spanK = X, there is a point of Gâteaux differentiability of f in x0 +K. This extends a Klee’s result for separable spaces. The well-known Mazur’s theorem says that a continuous conv...

We are interested in the question when a Banach space X with an unconditional basis is isomorphic (as a Banach space) to an order-continuous nonatomic Banach lattice. We show that this is the case if and only if X is isomorphic as a Banach space with X(i2). This and results of Bourgain are used to show that spaces HI (Tn) are not isomorphic to nonatomic Banach lattices. We also show that tent s...

Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the following properties: (1) B possesses an l norm in the sense that B is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on B and are representable through a bilinear form with a...

Each lower semi-continuous proper convex function f on a Banach space E defines a certain multivalued mapping of from E to E* called the subdifferential of f. It is shown here that the mappings arising this way are precisely the ones whose graphs are maximal "cyclically monotone" relations on Ex E*, and that each of these is also a maximal monotone relation. Furthermore, it is proved that of de...