We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael’s Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse o...

In this paper, we compare several Cesàro- and Kreiss-type boundedness conditions for a \(C_0\)-semigroup on Banach space show that those are all equivalent positive semigroup lattice. Furthermore, give an estimate of the growth rate Kreiss bounded eventually \((T_t)_{t\ge 0}\) certain lattices X. We prove if X is \(L^p\)-space, \(1<p<+\infty \), then \(\Vert T_t\Vert = \mathcal {O}\left( t/\log...

A net (xα) in a vector lattice X is unbounded order convergent to x ∈ X if |xα − x| ∧ u converges to 0 in order for all u ∈ X+. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A ne...

The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special class of Banach Lie algebras. The case of W -algebras is given particular attention. Semidirect products and the extension of the restricted Banach Lie-Poisson space by the Banach Lie-Poisson space of compact operators are given as examples.

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...

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...

in this paper, we introduce the cone normed spaces and cone bounded linear mappings. among other things, we prove the baire category theorem and the banach--steinhaus theorem in cone normed spaces.

The Haagerup norm ‖ · ‖h on the tensor product A ⊗ B of two C∗-algebras A and B is shown to be Banach space equivalent to either the Banach space projective norm ‖ · ‖γ or the operator space projective norm ‖ · ‖∧ if and only if either A or B is finite dimensional or A and B are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are ...

A survey is given of the work on strong regularity for uniform algebras over the last thirty years, and some new results are proved, including the following. Let A be a uniform algebra on a compact space X and let E be the set of all those points x ∈ X such that A is not strongly regular at x. If E has no non-empty, perfect subsets then A is normal, and X is the character space of A. If X is ei...

We analyze the construction of a sequence space Θ̃, resp. a sequence of sequence spaces, in order to have {gi} ∞ i=1 as a Θ̃-frame or Banach frame for a Banach space X , resp. pre-F -frame or F -frame for a Fréchet space XF = ∩s∈N0Xs, where {Xs}s∈N0 is a sequence of Banach spaces.