Bounded tightness for locally convex spaces and spaces C(X)

We show that metrizability and bounded tightness are actually equivalent for a large class of locally convex spaces including (LF)-spaces, (DF)-spaces, the space of distributions D′(Ω), etc. A consequence of this fact is that for the bounded tightness for the weak topology of X is equivalent to the following one: X is linearly homeomorphic to a subspace of . This nicely supplements very recent results of Cascales and Raja. Moreover, we show that a metric space X is separable if the space C_p(X) has bounded tightness.     

Generalized sampling in shift-invariant spaces with multiple stable generators

The aim of this paper is to derive stable generalized sampling in a shift-invariant space with ? stable generators. This is done in the light of the theory of frames in the product Hilbert space (? times). The generalized samples are expressed as the frame coefficients of an appropriate function in with respect to some particular frame in . Since any multiply stable generated shift-invariant space is the image of by means of a bounded invertible operator, the generalized sampling is obtained from some dual frame expansions in . An example in the setting of the Hermite cubic splines is exhibited.     

Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space whose dual space E^∗ satisfies the Kadec-Klee property, K be a closed convex nonempty subset of E. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . For arbitrary ?∈(0,1), let be a sequence in [?,1−?], for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space, K be a closed convex nonempty subset of E which is also a nonexpansive retract with retraction P. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . Let be a sequence in [?,1−?],?∈(0,1), for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

The unique minimal dual representation of a convex function

Suppose (i) X is a separable Banach space, (ii) C is a convex subset of X that is a Baire space (when endowed with the relative topology) such that aff(C) is dense in X, and (iii) f:C→R is locally Lipschitz continuous and convex. The Fenchel-Moreau duality can be stated as

     

Periodic solutions for a class of differential inclusions in general Banach spaces

Let X be a real Banach space, an m-accretive operator and a multi-function which is 2π-periodic with respect to its first argument, has nonempty, closed, convex and weakly compact values and is strongly-weakly upper semicontinuous. In this paper we prove the existence of at least one solution for the problem

     

A simple proof of inequalities of integrals of composite functions

In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continuous convex functions on a vector space Rm and vector-valued functions in a weakly compact subset of a Banach vector space generated by m-spaces for 1?p<+∞. Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by m-spaces instead.     

Additive Jordan isomorphisms of nest algebras on normed spaces

Let X be a real or complex Banach space. Let and be two nest algebras on X. Suppose that φ is an additive bijective mapping from onto such that φ(A²)=φ(A)² for every . Then φ is either a ring isomorphism or a ring anti-isomorphism. Moreover, if X is a real space or an infinite dimensional complex space, then there exists a continuous (conjugate) linear bijective mapping T such that either φ(A)=TAT⁻¹ for every or φ(A)=TA∗T−1 for every .     

Multipliers between Sobolev spaces and fractional differentiation

We characterize the pointwise multipliers which maps a Sobolev space to a Sobolev space in the case |s|<r<d/2.     

Generalized vector valued almost periodic and ergodic distributions

Schwartz's almost periodic distributions are generalized to the case of Banach space valued distributions , and furthermore for a given arbitrary class A to for φ∈ test functions D(R,C)}. It is shown that this extension process is iteration complete, i.e. . Moreover the T from are characterized in various ways, also tempered distributions with P={X-valued functions of polynomial growth} are shown. Under suitable assumptions , , where for all h>0}, is defined with the corresponding extension of Mh. With an extension of the indefinite integral from to D^′(R,X) a distribution analogue to the Bohl-Bohr-Amerio-Kadets theorem on the almost periodicity of bounded indefinite integrals of almost periodic functions is obtained, also for almost automorphic, Levitan almost periodic and recurrent functions, similar for a result of Levitan concerning ergodic indefinite integrals. For many of the above results a new (Δ)-condition is needed, we show that it holds for most of the A needed in applications. Also an application to the study of asymptotic behavior of distribution solutions of neutral integro-differential-difference systems is given.     

The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets

If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w^∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ₂-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w^∗-compact subset K⊂X∗∗ then and, if K∩C is w^∗-dense in K, then .     

Boundedness and unboundedness results for some maximal operators on functions of bounded variation

We characterize the space BV(I) of functions of bounded variation on an arbitrary interval I⊂R, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator MR from BV(I) into the Sobolev space W1,1(I). By restriction, the corresponding characterization holds for W1,1(I). We also show that if U is open in Rd, d>1, then boundedness from BV(U) into W1,1(U) fails for the local directional maximal operator , the local strong maximal operator , and the iterated local directional maximal operator . Nevertheless, if U satisfies a cone condition, then boundedly, and the same happens with , , and MR.     

On the intrinsic and the spatial numerical range

For a bounded function f from the unit sphere of a closed subspace X of a Banach space Y, we study when the closed convex hull of its spatial numerical range W(f) is equal to its intrinsic numerical range V(f). We show that for every infinite-dimensional Banach space X there is a superspace Y and a bounded linear operator such that . We also show that, up to renormig, for every non-reflexive Banach space Y, one can find a closed subspace X and a bounded linear operator T∈L(X,Y) such that .Finally, we introduce a sufficient condition for the closed convex hull of the spatial numerical range to be equal to the intrinsic numerical range, which we call the Bishop-Phelps-Bollobás property, and which is weaker than the uniform smoothness and the finite-dimensionality. We characterize strong subdifferentiability and uniform smoothness in terms of this property.     

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function and an absolutely summing operator u from X to another Banach space Y such that the composition is not Bochner integrable; in particular, h is not McShane integrable.     

On some geometric parameters in Banach spaces

Let X be a normed linear space and be the unit sphere of X. Let , , and J(X)=sup{‖x+y‖∧‖x−y‖}, x and y∈S(X) be the modulus of convexity, the modulus of smoothness, and the modulus of squareness of X, respectively. Let . In this paper we proved some sufficient conditions on δ(?), ρX(?), J(X), E(X), and , where the supremum is taken over all the weakly null sequence xn in X and all the elements x of X for the uniform normal structure.     

On the predictability of discrete dynamical systems III

Given an integer n?2, a metrizable compact topological n-manifold X with boundary and a finite positive Borel measure μ on X, we prove that “most” homeomorphisms are non-sensitive μ-almost everywhere on X. Moreover, we also prove that for “most” homeomorphisms the non-wandering set Ωf has μ-measure zero.     

Continuous dependence results for inhomogeneous ill-posed problems in Banach space

We apply semigroup theory and other operator-theoretic methods to prove Hölder-continuous dependence on modeling for the inhomogeneous ill-posed Cauchy problem in Banach space. The inhomogeneous ill-posed Cauchy problem is given by , u(0)=χ, 0?t<T; where −A is the infinitesimal generator of a holomorphic semigroup on a Banach space X, χ∈X, and . For a suitable function f, the approximate problem is given by , v(0)=χ. Under certain stabilizing conditions, we prove that for a related norm, where and M are computable constants independent of β, 0<β<1, and ω(t) is a harmonic function. These results extend earlier work of Ames and Hughes on the homogeneous ill-posed problem.     

Notes on ordered reproducing Hilbert spaces over the complex plane

Let f, g be entire functions. If there exist M₁,M₂>0 such that |f(z)|?M₁|g(z)| whenever |z|>M₂ we say that f?g. Let X be a reproducing Hilbert space with an orthogonal basis . We say that X is an ordered reproducing Hilbert space (or X is ordered) if f?g and g∈X imply f∈X. In this note, we show that if then X is ordered; if then X is not ordered. In the case , there are examples to show that X can be of order or opposite.