The rate of convergence for an almost surely convergent series of Banach space valued random elements is studied in this paper. As special cases of the main result, known results are obtained for a sequence of independent random elements in a Rademacher type p Banach space, and new results are obtained for a martingale difference sequence of random elements in a martingale type p Banach space a...

The Banach space `∞/c0 is isomorphic to the linear space of continuous functions on N∗ with the supremum norm, C(N∗). Similarly, the canonical representation of the `∞ sum of `∞/c0 is the Banach space of continuous functions on the closure of any non-compact cozero subset of N∗. It is important to determine if there is a continuous linear lifting of this Banach space to a complemented subset of...

We prove that there exists a Lipschitz function from l into IR which is Gâteaux-differentiable at every point and such that for every x, y ∈ l, the norm of f (x) − f (y) is bigger than 1. On the other hand, for every Lipschitz and Gâteaux-differentiable function from an arbitrary Banach space X into IR and for every ε > 0, there always exists two points x, y ∈ X such that ‖f (x)−f (y)‖ is less ...

where 0 < α≤ 1, Γ(α) is the gamma function, {A(t) : t ∈ [0,T]} is a family of linear closed operators defined on dense set D(A) in a Banach space E into E, u is the unknown Evalued function, u0 ∈ D(A), and f is a given E-valued function defined on [0,T]. It is assumed that D(A) is independent of t. Let B(E) denote the Banach space of all linear bounded operators in E endowed with the topology d...

Under consideration is the following question. If T is a linear operator from a normed function space 3 into a Banach space X, then may an appropriate topology 3 be defined so that conditions on T may be interpreted in terms of the topology 3? For the present we consider 3 to be the class of Banach function spaces. We define appropriate "semi-norm topologies" on the unit ball x! of the dual of ...

Abstract In this article, first, we present an example of fuzzy normed space by means the Mittag-Leffler function. Next, extend concept to matrix valued and also introduce a class control functions stabilize nonlinear ϕ -Hadamard fractional Volterra integro-differential equation. sense, investigate Ulam–Hyers–Rassias stability for kind equations in Banach space. Finally, as application, using f...

Here we research the univariate quantitative approximation, ordinary and fractional, of Banach space valued continuous functions on a compact interval or all real line by quasi-interpolation neural network operators. These approximations are derived establishing Jackson type inequalities involving modulus continuity engaged function its high order derivative fractional derivatives. Our operator...