共查询到20条相似文献,搜索用时 46 毫秒
1.
Marian Nowak 《Indagationes Mathematicae》2009,20(1):151-403
Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let X be a real Banach space and let X* stand for the Banach dual of X. Let B(Σ, X) be the Banach space of Σ-totally measurable functions f: Ω → X, and let B(Σ, X)* and B(Σ, X)** denote the Banach dual and the Banach bidual of B(Σ, X) respectively. Let bvca(Σ, X*) denote the Banach space of all countably additive vector measures ν: Σ → X* of bounded variation. We prove a form of generalized Vitali-Hahn-Saks theorem saying that relative σ(bvca(Σ, X*), B(Σ, X))-sequential compactness in bvca(Σ, X*) implies uniform countable additivity. We derive that if X reflexive, then every relatively σ(B(Σ, X)*, B(Σ, X))-sequentially compact subset of B(Σ, X)c~ (= the σ-order continuous dual of B(Σ, X)) is relatively σ(B(Σ, X)*, B(Σ, X)**)-sequentially compact. As a consequence, we obtain a Grothendieck type theorem saying that σ(B(Σ, X)*, B(Σ, X))-convergent sequences in B(Σ, X)c~ are σ(B(Σ, X)*, B(Σ, X)**)-convergent. 相似文献
2.
Ball-covering property of Banach spaces 总被引:7,自引:0,他引:7
Lixin Cheng 《Israel Journal of Mathematics》2006,156(1):111-123
We consider the following question: For a Banach spaceX, how many closed balls not containing the origin can cover the sphere of the unit ball? This paper shows that: (1) IfX is smooth and with dimX=n<∞, in particular,X=R n,then the sphere can be covered byn+1 balls andn+1 is the smallest number of balls forming such a covering. (2) Let Λ be the set of all numbersr>0 satisfying: the unit sphere of every Banach spaceX admitting a ball-covering consisting of countably many balls not containing the origin with radii at mostr impliesX is separable. Then the exact upper bound of Λ is 1 and it cannot be attained. (3) IfX is a Gateaux differentiability space or a locally uniformly convex space, then the unit sphere admits such a countable ball-covering if and only ifX * isw *-separable. 相似文献
3.
Jie-Hua Mai 《Topology and its Applications》2011,158(16):2216-2220
Let X be a topological space, f:X→X be a continuous map, and Y be a compact, connected and closed subset of X. In this paper we show that, if the boundary X∂Y contains exactly one point v and f(v)∈Y, then Y contains a minimal set of f. 相似文献
4.
Let (Ω,
, μ) be a measure space,
a separable Banach space, and
* the space of all bounded conjugate linear functionals on
. Let f be a weak* summable positive B(
*)-valued function defined on Ω. The existence of a separable Hilbert space
, a weakly measurable B(
)-valued function Q satisfying the relation Q*(ω)Q(ω) = f(ω) is proved. This result is used to define the Hilbert space L2,f of square integrable operator-valued functions with respect to f. It is shown that for B+(
*)-valued measures, the concepts of weak*, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained. 相似文献
5.
M. Fabian 《Journal of Mathematical Analysis and Applications》2008,339(1):735-739
Let (X,‖⋅‖) be a reflexive Banach space with Kadec-Klee norm. Let f:X→(−∞,+∞] be a function which is either Lipschitzian or is proper, bounded below, and lower semi-continuous. Then f is supported from below by residually many parabolas opening downward, that is, the infimal convolution of ‖⋅‖2 and f is attained at residually many points of X. 相似文献
6.
Jian YuDingtao Peng Shuwen Xiang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(17):6326-6332
Let X be a nonempty, convex and compact subset of normed linear space E (respectively, let X be a nonempty, bounded, closed and convex subset of Banach space E and A be a nonempty, convex and compact subset of X) and f:X×X→R be a given function, the uniqueness of equilibrium point for equilibrium problem which is to find x∗∈X (respectively, x∗∈A) such that f(x∗,y)≥0 for all y∈X (respectively, f(x∗,y)≥0 for all y∈A) is studied with varying f (respectively, with both varying f and varying A). The results show that most of equilibrium problems (in the sense of Baire category) have unique equilibrium point. 相似文献
7.
V. I. Rybakov 《Mathematical Notes》1996,60(2):175-185
Letf be a bounded Pettis integrable function ranging in a Banach spaceX (the range of the indefinite Pettis integral is separable). We consider Pettis integrability conditions for the Stone transform off and relate this problem to the regular oscillation condition for the family of functions {x
*
fx*B(X*)}, whereB(X*) is the unit ball inX
*.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 238–253, August, 1996. 相似文献
8.
B. Cascales 《Journal of Functional Analysis》2009,256(3):673-699
Kuratowski and Ryll-Nardzewski's theorem about the existence of measurable selectors for multi-functions is one of the keystones for the study of set-valued integration; one of the drawbacks of this result is that separability is always required for the range space. In this paper we study Pettis integrability for multi-functions and we obtain a Kuratowski and Ryll-Nardzewski's type selection theorem without the requirement of separability for the range space. Being more precise, we show that any Pettis integrable multi-function F:Ω→cwk(X) defined in a complete finite measure space (Ω,Σ,μ) with values in the family cwk(X) of all non-empty convex weakly compact subsets of a general (non-necessarily separable) Banach space X always admits Pettis integrable selectors and that, moreover, for each A∈Σ the Pettis integral coincides with the closure of the set of integrals over A of all Pettis integrable selectors of F. As a consequence we prove that if X is reflexive then every scalarly measurable multi-function F:Ω→cwk(X) admits scalarly measurable selectors; the latter is also proved when (X∗,w∗) is angelic and has density character at most ω1. In each of these two situations the Pettis integrability of a multi-function F:Ω→cwk(X) is equivalent to the uniform integrability of the family . Results about norm-Borel measurable selectors for multi-functions satisfying stronger measurability properties but without the classical requirement of the range Banach space being separable are also obtained. 相似文献
9.
Let (Ω, , μ) be a measure space, a separable Banach space, and 1 the space of all bounded conjugate linear functionals on . Let f be a weak1 summable positive B(1)-valued function defined on Ω. The existence of a separable Hilbert space , a weakly measurable B()-valued function Q satisfying the relation is proved. This result is used to define the Hilbert space L2,f of square integrable operator-valued functions with respect to f. It is shown that for B+(1)-valued measures, the concepts of weak1, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained. 相似文献
10.
It is shown that, given a lower semicontinuous convex integrandf satisfying a suitable integrability condition, there exists a sequence of Lipschitz simple integrands which Mosco converges tof and such that the sequence of conjugate integrands Mosco converges tof
*. Moreover, this sequence can be chosen so that the sequence of associated integral functionals, respectively defined onL
1(X) andL
(X
*), Mosco converges as well.We wish to thank Professor Erik J. Balder for interesting remarks on the first version of this work. 相似文献
11.
周云华 《数学物理学报(B辑英文版)》2011,31(1):102-108
Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space. 相似文献
12.
《Quaestiones Mathematicae》2013,36(2):185-214
Abstract We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds. 相似文献
13.
Heng Liu Lidong Wang Zhenyan Chu 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):6144-6147
Let X be a complete metric space without isolated points, and let f:X→X be a continuous map. In this paper we prove that if f is transitive and has a periodic point of period p, then f is distributionally chaotic in a sequence. Particularly, chaos in the sense of Devaney is stronger than distributional chaos in a sequence. 相似文献
14.
D. Azagra J.B. Seoane-Sepúlveda 《Journal of Mathematical Analysis and Applications》2009,354(1):229-233
If f is continuous on the interval [a,b], g is Riemann integrable (resp. Lebesgue measurable) on the interval [α,β] and g([α,β])⊂[a,b], then f○g is Riemann integrable (resp. measurable) on [α,β]. A well-known fact, on the other hand, states that f○g might not be Riemann integrable (resp. measurable) when f is Riemann integrable (resp. measurable) and g is continuous. If c stands for the continuum, in this paper we construct a c2-dimensional space V and a c-dimensional space W of, respectively, Riemann integrable functions and continuous functions such that, for every f∈V?{0} and g∈W?{0}, f○g is not Riemann integrable, showing that nice properties (such as continuity or Riemann integrability) can be lost, in a linear fashion, via the composite function. Similarly we construct a c-dimensional space W of continuous functions such that for every g∈W?{0} there exists a c-dimensional space V of measurable functions such that f○g is not measurable for all f∈V?{0}. 相似文献
15.
Nicholas J. Kuhn 《Advances in Mathematics》2006,201(2):318-378
Let K(n) be the nth Morava K-theory at a prime p, and let T(n) be the telescope of a vn-self map of a finite complex of type n. In this paper we study the K(n)*-homology of Ω∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PX be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map
sn(X):LT(n)P(X)→LT(n)Σ∞(Ω∞X)+ 相似文献
16.
Yingxuan Niu 《Journal of Mathematical Analysis and Applications》2011,376(2):528-534
Let X be a compact metric space and f:X→X be a continuous map. In this paper, we prove that if f has the average-shadowing property and the minimal points of f are dense in X, then f is weakly mixing and totally strongly ergodic. As applications we obtain that if f is a distal or Lyapunov stable map having the average-shadowing property, then X is consisting of one point. Moreover, we illustrate that the full shift has the average-shadowing property. 相似文献
17.
Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X * there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$ Then every continuous convex function g with $g \leqslant f$ on X is generically Fréchet differentiable. If, in addition, $\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty$ , then X is an Asplund space. 相似文献
18.
Let M be a smooth compact manifold and Λ be a compact invariant set.In this article,we prove that,for every robustly transitive set Λ,f|Λ satisfies a C1-genericstable shadowable property (resp.,C1-gene... 相似文献
19.
E. Jabłońska 《Acta Mathematica Hungarica》2009,125(1-2):113-119
Let n be a positive integer. We characterize solutions f: X → ? of the equation f (x + f(x) n y = f(x)f(y) mapping a real separable F-space X into ?, which are bounded on nonzero Christensen measurable sets. 相似文献
20.
Risong Li 《Communications in Nonlinear Science & Numerical Simulation》2012,17(7):2815-2823
Let f : X → X be a continuous map of a compact metric space X. The map f induces in a natural way a map fM on the space M(X) of probability measures on X, and a transformation fK on the space K(X) of closed subsets of X. In this paper, we show that if (X, f) is a chain transitive system with shadowing property, then exactly one of the following two statements holds:
- (a)
- fn and (fK)n are syndetically sensitive for all n ? 1.
- (b)
- fn and (fK)n are equicontinuous for all n ? 1.