共查询到20条相似文献,搜索用时 15 毫秒
1.
Manuel González Antonio Martínez-Abejón 《Journal of Mathematical Analysis and Applications》2007,327(2):816-828
A local dual of a Banach space X is a closed subspace of X∗ that satisfies the properties that the principle of local reflexivity assigns to X as a subspace of X∗∗. Here we introduce a technical property which characterizes the local dual spaces of a Banach space and allows us to show new examples of local dual spaces. 相似文献
2.
Ju Myung Kim 《Journal of Mathematical Analysis and Applications》2006,324(1):721-727
It is shown that for the separable dual X∗ of a Banach space X, if X∗ has the weak approximation property, then X∗ has the metric weak approximation property. We introduce the properties W∗D and MW∗D for Banach spaces. Suppose that M is a closed subspace of a Banach space X such that M⊥ is complemented in the dual space X∗, where for all m∈M}. Then it is shown that if a Banach space X has the weak approximation property and W∗D (respectively, metric weak approximation property and MW∗D), then M has the weak approximation property (respectively, bounded weak approximation property). 相似文献
3.
We study the Pettis integral for multi-functions defined on a complete probability space (Ω,Σ,μ) with values into the family cwk(X) of all convex weakly compact non-empty subsets of a separable Banach space X. From the notion of Pettis integrability for such an F studied in the literature one readily infers that if we embed cwk(X) into ?∞(BX∗) by means of the mapping defined by j(C)(x∗)=sup(x∗(C)), then j○F is integrable with respect to a norming subset of B?∞∗(BX∗). A natural question arises: When is j○F Pettis integrable? In this paper we answer this question by proving that the Pettis integrability of any cwk(X)-valued function F is equivalent to the Pettis integrability of j○F if and only if X has the Schur property that is shown to be equivalent to the fact that cwk(X) is separable when endowed with the Hausdorff distance. We complete the paper with some sufficient conditions (involving stability in Talagrand's sense) that ensure the Pettis integrability of j○F for a given Pettis integrable cwk(X)-valued function F. 相似文献
4.
Sergey V. Astashkin 《Journal of Functional Analysis》2011,260(1):195-207
Let X be a rearrangement invariant function space on [0,1]. We consider the subspace Radi X of X which consists of all functions of the form , where xk are arbitrary independent functions from X and rk are usual Rademacher functions independent of {xk}. We prove that Radi X is complemented in X if and only if both X and its Köthe dual space X′ possess the so-called Kruglov property. As a consequence we show that the last conditions guarantee that X is isomorphic to some rearrangement invariant function space on [0,∞). This strengthens earlier results derived in different approach in [W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 1 (217) (1979)]. 相似文献
5.
We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y,X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y,X) for X a classical Banach space such as ?p,Lp,L1,C(ωω) or C[0,1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c0 or ?2? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y,X)=1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L∞-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X∗ are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c0 or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L∞ or a superreflexive type 2 Banach lattice. 相似文献
6.
T.S.S.R.K. Rao 《Journal of Mathematical Analysis and Applications》2007,328(2):1173-1177
In this paper we consider proximinality questions for higher ordered dual spaces. We show that for a finite dimensional uniformly convex space X, the space C(K,X) is proximinal in all the duals of even order. For any family of uniformly convex Banach spaces {Xα}{α∈Γ} we show that any finite co-dimensional proximinal subspace of X=c0⊕Xα is strongly proximinal in all the duals of even order of X. 相似文献
7.
By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X∗ of X is w∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?1[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0. 相似文献
8.
Ya.I. Alber 《Journal of Mathematical Analysis and Applications》2005,312(1):330-342
We establish decompositions of a uniformly convex and uniformly smooth Banach space B and dual space B∗ in the form B=M?J∗M⊥ and B∗=M⊥?JM, where M is an arbitrary subspace in B, M⊥ is its annihilator (subspace) in B∗, J:B→B∗ and J∗:B∗→B are normalized duality mappings. The sign ? denotes the James orthogonal summation (in fact, it is the direct sums of the corresponding subspaces and manifolds). In a Hilbert space H, these representations coincide with the classical decomposition in a shape of direct sum of the subspace M and its orthogonal complement M⊥: H=M⊕M⊥. 相似文献
9.
Changsun Choi 《Journal of Mathematical Analysis and Applications》2006,323(1):78-87
We introduce the properties W∗D and BW∗D for the dual space of a Banach space. And then solve the dual problem for the compact approximation property (CAP): if X∗ has the CAP and the W∗D, then X has the CAP. Also, we solve the three space problem for the CAP: for example, if M is a closed subspace of a Banach space such that M⊥ is complemented in X∗ and X∗ has the W∗D, then X has the CAP whenever X/M has the CAP and M has the bounded CAP. Corresponding problems for the bounded compact approximation property are also addressed. 相似文献
10.
Hyun Ho Lee 《Journal of Functional Analysis》2011,260(1):135-145
In this paper we generalize the notion of essential codimension of Brown, Douglas, and Fillmore using KK-theory and prove a result which asserts that there is a unitary of the form ‘identity + compact’ which gives the unitary equivalence of two projections if the ‘essential codimension’ of two projections vanishes for certain C∗-algebras employing the proper asymptotic unitary equivalence of KK-theory found by M. Dadarlat and S. Eilers. We also apply our result to study the projections in the corona algebra of C(X)⊗B where X is [0,1], (−∞,∞), [0,∞), and [0,1]/{0,1}. 相似文献
11.
M. Raja 《Journal of Mathematical Analysis and Applications》2004,290(1):63-75
Given an injective bounded linear operator T:X→Y between Banach spaces, we study the Borel measurability of the inverse map T−1:TX→X. A remarkable result of Saint-Raymond (Ann. Inst. Fourier (Grenoble) 26 (1976) 211-256) states that if X is separable, then the Borel class of T−1 is α if, and only if, X∗ is the αth iterated sequential weak∗-closure of T∗Y∗ for some countable ordinal α. We show that Saint-Raymond's result holds with minor changes for arbitrary Banach spaces if we assume that T has certain property named co-σ-discreteness after Hansell (Proc. London Math. Soc. 28 (1974) 683-699). As an application, we show that the Borel class of the inverse of a co-σ-discrete operator T can be estimated by the image of the unit ball or the restrictions of T to separable subspaces of X. Our results apply naturally when X is a WCD Banach space since in this case any injective bounded linear operator defined on X is automatically co-σ-discrete. 相似文献
12.
Ken-Ichi Mitani 《Journal of Mathematical Analysis and Applications》2007,327(2):898-907
Let X be a Banach space and ψ a continuous convex function on [0,1] satisfying certain conditions. Let Xψ⊕X be the ψ-direct sum of X. In this note, we characterize the strict convexity, uniform convexity and uniformly non-squareness of Banach spaces using ψ-direct sums, which extends the well-known characterization of these spaces. 相似文献
13.
Let X be a Banach space and α ∈ (0, 1]. We find equivalent conditions for a function f: [0,1] → X to admit an equivalent parametrization, which is C 1,α (i.e., has α-Hölder derivative). For X = ?, a characterization is well-known. However, even in the case X = ?2 several new ideas are needed. 相似文献
14.
Serguei V. Astashkin 《Journal of Functional Analysis》2009,256(12):4071-4094
Let X be a rearrangement invariant function space on [0,1]. We consider the Rademacher multiplicator space Λ(R,X) of all measurable functions x such that x⋅h∈X for every a.e. converging series h=∑anrn∈X, where (rn) are the Rademacher functions. We study the situation when Λ(R,X) is a rearrangement invariant space different from L∞. Particular attention is given to the case when X is an interpolation space between the Lorentz space Λ(φ) and the Marcinkiewicz space M(φ). Consequences are derived regarding the behaviour of partial sums and tails of Rademacher series in function spaces. 相似文献
15.
Shangquan Bu 《Journal of Mathematical Analysis and Applications》2003,288(1):246-250
If A is a sectorial operator on a Banach space X, then the space C([0,1];(X,D(A))θ,∞) is a subspace of the interpolation space (C([0,1];X),C([0,1];D(A)))θ,∞. The inclusion is strict in general. 相似文献
16.
Mariusz Urbański 《Topology and its Applications》2009,156(17):2762-2771
Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X)?ω0, then HD(X)=+∞. We prove that tHD(X)?ω1 for every separable metric space X, and, as our main theorem, we show that for every ordinal number α<ω1 there exists a compact metric space Xα (a subspace of the Hilbert space l2) with tHD(Xα)=α and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension. 相似文献
17.
Spiros A. Argyros 《Advances in Mathematics》2007,209(2):666-748
We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C([0,1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L1(0,1) such that the indices β and rND are unbounded on the set of Baire-1 elements of the ball of the double dual R∗∗ of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y∈C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X. 相似文献
18.
S. V. Astashkin 《Functional Analysis and Its Applications》2013,47(2):148-151
We show that, for a broad class of symmetric spaces on [0, 1], the complementability of the subspace generated by independent functions f k (k = 1, 2,…) is equivalent to the complementability of the subspace generated by the disjoint translates $\bar f_k (t) = f_k (t - k + 1)\chi _{[k - 1,k]} (t)$ of these functions in some symmetric space Z X 2 on the semiaxis [0,∞). Moreover, if Σ k=1 ∞ m(supp f k ) ? 1, then Z X 2 can be replaced by X itself. This result is new even in the case of L p -spaces. A series of consequences is obtained; in particular, for the class of symmetric spaces, a result similar to a well-known theorem of Dor and Starbird on the complementability in L p [0, 1] (1 ? p < ∞) of the subspace [f k ] generated by independent functions provided that it is isomorphic to the space l p is obtained. 相似文献
19.
The two main results are:
- A.
- If a Banach space X is complementably universal for all subspaces of c0 which have the bounded approximation property, then X∗ is non-separable (and hence X does not embed into c0).
- B.
- There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.
20.
Christian Le Merdy 《Advances in Mathematics》2010,224(4):1641-2998
Let G be an amenable group, let X be a Banach space and let π:G→B(X) be a bounded representation. We show that if the set is γ-bounded then π extends to a bounded homomorphism w:C∗(G)→B(X) on the group C∗-algebra of G. Moreover w is necessarily γ-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G=Z, R or T, and/or when X has property (α). 相似文献