Effective constructions of separable quotients of Banach spaces

Acta Mathematica Hungarica, 2015

It is shown that an infinite dimensional Banach space is hyperbarrelled if and only if it does not admit an infinite dimensional separable quotient.

Central European Journal of Mathematics, 2011

Let X , Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R : X → Y such that its range, R(X ), is dense in Y .

Transactions of the American Mathematical Society, 1987

ABSTRACT. For a topological space X, let Ci(X) denote the Banach space of all bounded functions /: X —► R such that for every e > 0 the set {x € X: \f{x)\ > e} is closed and discrete in X, endowed with the supremum norm. The main theorem is the following: Let L be a weakly countably ...

Revista Matemática Complutense, 2020

We show that c 0 is not a projective Banach lattice, answering a question of B. de Pagter and A. Wickstead. .On the other hand, we show that c 0 is complemented in the free Banach lattice generated by itself (seen as a Banach space). As a consequence, the free Banach lattice generated by c 0 is not projective.

We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y , then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space X has the bounded approximation property and Y is Lipschitz isomorphic to X, then Y has the bounded approximation property.

Studia Mathematica, 2008

We show that an infinite-dimensional complete linear space X has: • a dense hereditarily Baire Hamel basis if |X| ≤ c + ; • a dense non-meager Hamel basis if |X| = κ ω = 2 κ for some cardinal κ.

Israel Journal of Mathematics, 1978

A. Pelczynski [3] and H. P. Rosenthal [5] considered conditions on a bounded linear operator T from C{K), K compact metric, into an arbitrary Banach space which ensure that T is an isomorphism on a subspace of C(K) isomorphic to a C(S) space. (Throughout T will denote a bounded linear operator from C(K), K compact metric, into a Banach space X.) Both used conditions on the "size" of T*B X * to produce their results. Pefczynski showed that if T is a nonweakly compact operator there is a subspace Y of C(K) such that Y is isometric to c 0 and the restriction of T to Y is an isomorphism. Rosenthal's result is similar in form: If T*B X * is nonseparable then there is a subspace Y of C(K) such that Y is isometric to C(A) (A is the Cantor set) and the restriction of T to Y is an isomorphism. Here we announce a similar result for C 0 (co w), the space of continuous functions on the ordinals not greater than co w and vanishing at co w (the ordinals are considered in the order topology). To state our result we need to recall the notion of index of a Banach space introduced by Szlenk [6]. DEFINITION. Let A be a bounded subset of a separable Banach space X and B be a bounded w*-closed subset of X*. For each e > 0 let P 0 (e, A> B) =B and define inductively a family of subsets of B indexed by the countable ordinals as follows: For each ordinal a < co r /> a+1 = {b\3(b n) C P a (e, A, B), b n ~^ b, and 3(# w) C A, a n-^ 0, such that hm n^00 (b n , a n) > e} and if j3 is a limit ordinal, let P^(e, A, B) = f) a< pP a (e, A, B). The e-Szlenk index of A and B is r?(e, A, B) = sup {a: P a (e, A, B)*0}. Before we state our theorem let us note that since T is nonweakly compact if and only if r?(e, B c^K y T*B X *) > 1 for some e > 0, we can restate Pelczynskf s theorem in terms of the index: If for some e > 0, r?(e, B c^K y T*B X *) > 1, there is a subspace Y of C(K) such that Y is isometric to c 0 and the restriction of T to Y is an isomorphism. In this form our results are natural extensions of Petczynski's. THEOREM 1. For every k EZ + and e > 0 there is an integer n(e, k) such AMS (MOS) subject classifications (1970). Primary 46E15.

2011

We exhibit examples of Fréchet Montel spaces E which have a nonreexive Fréchet quotient but such that every Banach quotient is nite dimensional. The construction uses a method developed by Albanese and Moscatelli and requires new ingredients. Some of the main steps in the proof are presented in Section 2. They are of independent interest and show for example that the canonical inclusion between James spaces Jp ⊂ Jq, 1 < p < q < ∞, is strictly cosingular. This result requires a careful analysis of the block basic sequences of the canonical basis of the dual J p of the James space J p , and permits us to show that the Fréchet space J p + = ∩q>pJq has no innitedimensional Banach quotients. Plichko and Maslyuchenko had proved that it has no innitedimensional Banach subspaces.

Bulletin of the American Mathematical Society, 1976

Arxiv preprint math/0406479, 2004

Abstract: We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder-Bernstein Index of any of these spaces is $2^{\ aleph_0} $. Our construction is based on a Banach space ...