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Register a new userBanach Spaces of Bounded Szlenk Index II
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For every $alpha<omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $omega^{alphaomega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $omega^{alphaomega}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates.
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We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (non-complete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index~1). On the other hand, we show that a lush real Banach space is neither strictly convex nor smooth, unless it is one-dimensional. In particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of any lush infinite-dimensional real space contains a copy of $ell_1$.
We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.
We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $ell_1$ and that operators which do not fix copies of $ell_1$ on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.
Assume that $mathcal{I}$ is an ideal on $mathbb{N}$, and $sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(mathcal{I}):=left{t in {0,1}^{mathbb{N}} colon sum_n t(n)x_n textrm{ is } mathcal{I}textrm{-convergent}right}$. In the category case, we assume that $mathcal{I}$ has the Baire property and $sum_n x_n$ is not unconditionally convergent, and we deduce that $A(mathcal{I})$ is meager. We also study the smallness of $A(mathcal{I})$ in the measure case when the Haar probability measure $lambda$ on ${0,1}^{mathbb{N}}$ is considered. If $mathcal{I}$ is analytic or coanalytic, and $sum_n x_n$ is $mathcal{I}$-divergent, then $lambda(A(mathcal{I}))=0$ which extends the theorem of Dindov{s}, v{S}alat and Toma. Generalizing one of their examples, we show that, for every ideal $mathcal{I}$ on $mathbb{N}$, with the property of long intervals, there is a divergent series of reals such that $lambda(A(Fin))=0$ and $lambda(A(mathcal{I}))=1$.
The purpose of this article is to present the construction and basic properties of the general Bochner integral. The approach presented here is based on the ideas from the book The Bochner Integral by J. Mikusinski where the integral is presented for functions defined on $mathbb{R}^N$. In this article we present a more general and simplified construction of the Bochner integral on abstract measure spaces. An extension of the construction to functions with values in a locally convex space is also considered.
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