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1.
Let X be an infinite-dimensional Banach space with weight τ. By Cld
AW
(X), we denote the hyperspace of nonempty closed sets in X with the Attouch—Wets topology. By Fin
AW
(X), Comp
AW
(X) and Bdd
AW
(X), we denote the subspaces of Cld
AW
(X) consisting of finite sets, compact sets and bounded closed sets, respectively. In this paper, it is proved that
Fin
AW
(X)≈Comp
AW
(X)≈ℓ2(τ)×ℓ2
f
ℓandℓBdd
AW
(X)≈ℓ2(2τ)×ℓ2
f
where ≈ means ‘is homeomorphic to’, ℓ2(τ) is the Hilbert space with weight τ (ℓ2(ℵ0)=ℓ2 the separable Hilbert space) and
ℓ2
f
={(x
i
)
iεN
εℓ2∣x
i
=0 except for finitely many iεN}. 相似文献
2.
Sebastian Król 《Semigroup Forum》2009,79(2):369-376
We show that every contractive C
0-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C
0-groups in the weak operator topology uniformly on compact subsets of ℝ+. As a consequence we get a new characterization of a bounded H
∞-calculus for the negatives of generators of bounded holomorphic semigroups. Applications of our results to the study of a
topological structure of the set of (almost) weakly stable contractive C
0-semigroups on X are also discussed.
The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW
grant Nr. N201384834. 相似文献
3.
Let X denote a specific space of the class of X
α,p
Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ℓp Banach spaces. We show that for p > 1 the Banach space X contains asymptotically isometric copies of ℓp. It is known that any member of the class is a dual space. We show that the predual of X contains isometric copies of ℓp where 1/p + 1/q = 1. For p = 1 it is known that the predual of the Banach space X contains asymptotically isometric copies of c
0. Here we give a direct proof of the known result that X contains asymptotically isometric copies of ℓ1. 相似文献
4.
Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover of X there is a sequence of maps (f
n
: X → X)
nεgw
such that each f
n
is -near to the identity map of X and the family {f
n
(X)}
n∈ω
is locally finite in X. Also we show that a metrizable space X of density dens(X) < is a Hilbert manifold if X has gw-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z
∞-set in X.
相似文献
5.
If (Ω,Σ) is a measurable space and X a Banach space, we provide sufficient conditions on Σ and X in order to guarantee that bvca(Σ, X) the Banach space of all X-valued countably additive measures of bounded variation equipped with the variation norm, contains a copy of c0 if and only if X does.
This work was supported by the project MTM2005-01182 of the Spanish Ministry of Education and Science, co-financed by the
European Community (Feder projects). 相似文献
6.
D. P. Bertsekas 《Journal of Optimization Theory and Applications》2008,139(2):209-225
We consider the problem
where x
i
are multidimensional subvectors of x, f
i
are convex functions, and S is a subspace. Monotropic programming, extensively studied by Rockafellar, is the special case where the subvectors x
i
are the scalar components of x. We show a strong duality result that parallels Rockafellar’s result for monotropic programming, and contains other known
and new results as special cases. The proof is based on the use of ε-subdifferentials and the ε-descent method, which is used here as an analytical vehicle.
Work partially supported by the National Science Foundation Grant No. CCR-9731273. 相似文献
7.
MiaoLI QuanZHENG 《数学学报(英文版)》2004,20(5):821-828
Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups. 相似文献
8.
Given separable Banach spaces X, Y, Z and a bounded linear operator T:X→Y, then T is said to preserve a copy of Z provided that there exists a closed linear subspace E of X isomorphic to Z and such that the restriction of T to E is an into isomorphism. It is proved that every operator on C([0,1]) which preserves a copy of an asymptotic ℓ1 space also preserves a copy of C([0,1]). 相似文献
9.
Dong Hyun Cho 《Czechoslovak Mathematical Journal》2009,59(2):431-452
Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w
ϕ of Wiener measure and for a partition 0 = t
0 < t
1 < ... < t
n
< t
n+1 = T of [0, T], let X
n
: C[0, T] → ℝ
n+1 and X
n+1: C[0, T] → ℝ
n+2 be given by X
n
(x) = (x(t
0), x(t
1), ..., x(t
n
)) and X
n+1(x) = (x(t
0), x(t
1), ..., x(t
n+1)), respectively.
In this paper, using a simple formula for the conditional w
ϕ-integral of functions on C[0, T] with the conditioning function X
n+1, we derive a simple formula for the conditional w
ϕ-integral of the functions with the conditioning function X
n
. As applications of the formula with the function X
n
, we evaluate the conditional w
ϕ-integral of the functions of the form F
m
(x) = ∫0
T
(x(t))
m
for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X
n
, we evaluate the conditional w
ϕ-integral of the functions in a Banach algebra
which is an analogue of the Cameron and Storvick’s Banach algebra
. Finally, we derive the conditional analytic Feynman w
ϕ-integrals of the functions in
.
相似文献
10.
A closed, convex and bounded setP in a Banach spaceE is called a polytope if every finite-dimensional section ofP is a polytope. A Banach spaceE is called polyhedral ifE has an equivalent norm such that its unit ball is a polytope. We prove here:
We deduce from these two results that in a polyhedral Banach space (for instance in c0(ℕ) or inC(K) forK countable compact), every equivalent norm can be approximated by norms which are analytic onE/{0}. 相似文献
(1) | LetW be an arbitrary closed, convex and bounded body in a separable polyhedral Banach spaceE and let ε>0. Then there exists a tangential ε-approximating polytopeP for the bodyW. |
(2) | LetP be a polytope in a separable Banach spaceE. Then, for every ε>0,P can be ε-approximated by an analytic, closed, convex and bounded bodyV. |
11.
S. M. Bates 《Israel Journal of Mathematics》1997,100(1):209-220
It is shown that (1) every infinite-dimensional Banach space admits aC
1 Lipschitz map onto any separable Banach space, and (2) if the dual of a separable Banach spaceX contains a normalized, weakly null Banach-Saks sequence, thenX admits aC
∞ map onto any separable Banach space. Subsequently, we generalize these results to mappings onto larger target spaces.
Supported by an NSF Postdoctoral Fellowship in Mathematics. 相似文献
12.
We prove that there exists a Lipschitz function froml
1 into ℝ2 which is Gateaux-differentiable at every point and such that for everyx, y εl
1, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gateaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gateaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0.
This work has been initiated while the second-named author was visiting the University of Bordeaux. The second-named author
is supported by grant AV 1019003, A1 019 205, GA CR 201 01 1198. 相似文献
13.
We study in this paper an M/M/1 queue whose server rate depends upon the state of an independent Ornstein–Uhlenbeck diffusion process (X(t)) so that its value at time t is μ
φ(X(t)), where φ(x) is some bounded function and μ>0. We first establish the differential system for the conditional probability density functions of the couple (L(t),X(t)) in the stationary regime, where L(t) is the number of customers in the system at time t. By assuming that φ(x) is defined by φ(x)=1−ε((x
∧
a/ε)∨(−b/ε)) for some positive real numbers a, b and ε, we show that the above differential system has a unique solution under some condition on a and b. We then show that this solution is close, in some appropriate sense, to the solution to the differential system obtained
when φ is replaced with Φ(x)=1−ε
x for sufficiently small ε. We finally perform a perturbation analysis of this latter solution for small ε. This allows us to check at the first order the validity of the so-called reduced service rate approximation, stating that
everything happens as if the server rate were constant and equal to
.
相似文献
14.
We give sufficient conditions on Banach spaces X and Y so that their projective tensor product X ⊗π
Y, their injective tensor product X ⊗ɛ
Y, or the dual (X ⊗π
Y)* contain complemented copies of ℓp. 相似文献
15.
J. Borsík 《Acta Mathematica Hungarica》2007,115(4):319-332
Let X be a topological space and (Y,d) be a metric space. If f: X → Y is a function then there is a function a
f
: X → [0, ∞] such that f is almost continuous at x if and only if a
f
(x) = 0. Some properties of this function are investigated.
Supported by grant VEGA 2/6087/26 and APVT-51-006904. 相似文献
16.
Petr Hájek 《Journal of Differential Equations》2010,249(12):3342-3351
We show that if X is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping f:X→X such that the autonomous differential equation x′=f(x) has no solution at any point. 相似文献
17.
A. G. Baskakov 《Mathematical Notes》2000,67(6):690-698
We obtain conditions for the invertibility and the Fredholm property of the difference operator (Dx)(n)=x(n) -U(n)x(n − 1),n ε ℤ, in the Banach space l
p
(ℤ, X),p ε [1, ∞], of vector sequences, whereX is a Banach space andU is a bounded operator function.
Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 816–827, June, 2000. 相似文献
18.
J.C. FERRANDO S. MOLL 《数学学报(英文版)》2007,23(9):1593-1600
In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc (X) has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc (X) contains a dense (LM) subspace and if and only if X is a-compact. 相似文献
19.
《Quaestiones Mathematicae》2013,36(1):87-98
An absolutely representing system (ARS) in a Banach space X is a set D ? X such that every vector x in X admits a representation by an absolutely convergent series x = Σ i a i x i with (a i ) ? R and (x i ) ? D. We investigate some general properties of absolutely representing systems. In particular, absolutely representing systems in uniformly smooth and in B-convex Banach spaces are characterized via ?-nets of the unit balls. Every absolutely representing system in a B-convex Banach space is quick, i.e., in the representation above one can achieve ∥a i x i ∥ < cq i ∥x∥, i = 1, 2,… for some constants c > 0 and q ? (0,1). 相似文献
20.
We introduce a notion which is intermediate between that of taking thew*-closed convex hull of a set and taking the norm closed convex hull of this set. This notion helps to streamline the proof
(given in [FLP]) of the famous result of James in the separable case. More importantly, it leads to stronger results in the
same direction. For example:
相似文献
1. | AssumeX is separable and non-reflexive and its unit sphere is covered by a sequence of balls of radiusa<1. Then for every sequence of positive numbers tending to 0 there is anf εX*, such that ‖f‖ = 1 andf (x)≤1 −ε i , wheneverx εC i ,i=1,2,… |
2. | AssumeX is separable and non-reflexive and letT:Y →X* be a bounded linear non-surjective operator. Then there is anf εX* which does not attain its norm onB X such thatf ∉T(Y). |