Complemented copies of ℓ<Subscript>p</Subscript> spaces in tensor products

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 ℓ_p.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

Hyperspaces of Banach Spaces with the Attouch—Wets Topology

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(2^τ)×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}.     

Embedding c<Subscript>0</Subscript> in bvca(Σ, <Emphasis Type="Italic">X</Emphasis>)

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).     

Operators on <Emphasis Type="Italic">C</Emphasis>[0,1] preserving copies of asymptotic ℓ<Subscript>1</Subscript> spaces

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]).     

Further properties of Azimi-Hagler banach spaces

For the Azimi-Hagler spaces more geometric and topological properties are investigated. Any constructed space is denoted by X _α,p . We show

Coefficient Quantization in Banach Spaces

Let (e _i ) be a dictionary for a separable infinite-dimensional Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a ‘finite alphabet’. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c ₀. We also show that, for every ε>0, the unit ball of every separable infinite-dimensional Banach space X contains a dictionary (x _i ) such that the additive group generated by (x _i ) is (3+ε)⁻¹-separated and 1/3-dense in X.      

Containment of c<Subscript>0</Subscript> and ℓ<Subscript>1</Subscript> in π<Subscript>1</Subscript>(<Emphasis Type="Italic">E,F</Emphasis>)

Suppose π₁(E, F) is the space of all absolutely 1-summing operators between two Banach spacesE andF. We show that ifF has a copy of c₀, then π₁ (E, F) will have a copy of c₀, and under some conditions ifE has a copy of ℓ₁ then π₁ (E, F) would have a complemented copy of ℓ₁.     

Concerning Bourgain’s ℓ1-Index of a Banach space

A well known argument of James yields that if a Banach spaceX contains ℓ ₁ ⁿ ’s uniformly, thenX contains ℓ ₁ ⁿ ’s almost isometrically. In the first half of the paper we extend this idea to the ordinal ℓ₁-indices of Bourgain. In the second half we use our results to calculate the ℓ₁-index of certain Banach spaces. Furthermore we show that the ℓ₁-index of a separable Banach space not containing ℓ₁ must be of the form ω^α for some countable ordinal α. Research supported by the NSF and TARP.     

On a Banach space approximable by Jacobi polynomials

Let X represent either the space C[-1,1] L _p ^(α,β) (w), 1 ≦ p < ∞ on [-1, 1]. Then Xare Banach spaces under the sup or the p norms, respectively. We prove that there exists a normalized Banach subspace X ¹ _αβ of Xsuch that every f ∈ X ¹ _αβ can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Our method to prove such an approximation problem is Fourier–Jacobi analysis based on the convergence of Fourier–Jacobi expansions. This revised version was published online in June 2006 with corrections to the Cover Date.