Weak ∗-Polish Banach spaces

Letr, s ∈ [0, 1], and let X be a Banach space satisfying the M(r, s)-inequality, that is, $\parallel x^{***} \parallel \geqslant r\parallel \pi _X x^{***} \parallel + s\parallel x^{***} - \pi _X x^{***} \parallel for x^{***} \in X^{***} ,$ where πX is the canonical projection from X*** onto X*. We show some examples of Banach spaces not containing c0, having the point of continuity property and satisfying the above inequality for r not necessarily equal to one. On the other hand, we prove that a Banach space X satisfying the above inequality for s=1 admits an equivalent locally uniformly rotund norm whose dual norm is also locally uniformly rotund. If, in addition, X satisfies $\mathop {\lim \sup }\limits_\alpha \parallel u^* + sx_\alpha ^* \parallel \leqslant \mathop {\lim \sup }\limits_\alpha \parallel v^* + x_\alpha ^* \parallel $ whenever u*, v* ∈X* with ‖u*‖≤‖v*‖ and (x ${}_{α}^{*}$ ) is a bounded weak* null net in X*, then X can be renormed to satisfy the, M(r, 1) and the M(1, s)-inequality such that X* has the weak* asymptotic-norming property I with respect to BX.

Let E be a Banach space. The concept of n‐type overE is introduced here, generalizing the concept of type overE introduced by Krivine and Maurey. Let E″ be the second dual of E and fix . The function τ : E × ℝn → ℝ, defined by letting for all x ∈ E and all a1, …, an ∈ ℝ, defines an n‐type over E. Types that can be represented in this way are called double‐dual n‐types; we say that realizes τ. Let E be a (not necessarily separable) Banach space that does not contain ℓ1. We study the set of elements of that realize a given double‐dual n‐type over E. We show that the set of realizations of this n‐type is convex. This generalizes a result of Haydon and Maurey who showed that the set of realizations of a given 1‐type over a separable Banach space E is convex. The proof makes use of Henson′s language for normed space structures and uses ideas from mathematical logic, most notably the Löwenheim‐Skolem theorem.

In a recent paper, F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E), the Banach space of continuous functions from a compact Hausdorff space Q to E, has the Dieudonne property. They asked whether or not the result is still true if one only assumes that E does not contain a copy of l1. In this paper we give a positive answer to their question. As a corollary we show that if E is a subspace of an order continuous Banach lattice, then E has the Dieudonne property if and only if C(Q, E) has the same property. If E is a Banach space and Q is a compact Hausdorff space, then C(Q, E) will stand for the Banach space of the E-valued continuous functions on Q under the supremum norm. A Banach space E is said to have the Dieudonne property if for every Banach space F, any bounded linear operator T: E -> F that transforms weakly Cauchy sequences into weakly convergent sequences is weakly compact. In [3] F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E) has the Dieudonne property and they asked whether the same result is true when replacing the assumption that E* is separable by supposing only that 11 does not embed in E. In this paper we give a positive answer to their question. Recall that a topological space (X, -y) is said to be Polish if it is homeomorphic to a separable complete metric space and it is said to be analytic if it is the continuous image of a Polish space. A subset A of a topological space (X, y) is said to be coanalytic if its complement (X\A, y) is analytic. Finally A is said to be PCA if it is the continuous image of a coanalytic space. The notations and terminology used and not defined can be found in [5, 8, or 10]. In the proof of Lemma 3 we need the following two results. THEOREM 1 (M. SREBRNY [9]). Let X and Y be two analytic spaces and let F be a multivalued function from X to the subsets of Y, such that its graph is PCA and for which one can prove that for every x c X, F(x) :8 0 using only the axioms of ZFC. Then there exists a universally measurable map f: X -> Y such that f (x) c F(x) for every x E X. THEOREM 2 (I. ASSANI [1, 2]). Let E be a separable Banach space. The set of weakly Cauchy sequences is a coanalytic subset of EN. Received by the editors January 5, 1985. 1980 Mathematics Subject Classification. Primary 46G10, 46B22.

Every separable Banach space admits an equivalent norm such that the unit ball with respect to this norm has at most countably many strongly extreme points.Every separable nonreflexive Banach space can be renormed so that its unit ball has at most countably many weakly strongly extreme points.

On constructing multiplications on Banach spaces

The main purpose of this work is to apply elements of set theory to the study of biorthogonal systems in Banach spaces. Initially, we study Markushevic's Theorem, which ensures that every separable Banach space has a countable biorthogonal system. With this result, we focus our attention to the study of nonseparable Banach spaces, more especifically, we study the existence of uncountable biorthogonal systems in Banach spaces of the form C(K), with K a nonmetrizable compact Hausdorff space. In this direction, we study two theorems of S. Todorcevic. The first one gives us sufficient conditions that a compact Hausdorff space K must satisfy in order to get that the respective Banach space C(K) has an uncountable biorthogonal system. The second one tells us that under Martin's Axiom, every nonseparable Banach space of the form C(K) has an uncountable biorthogonal system. Next, we consider some cardinal functions defined by P. Koszmider for Banach spaces, related with biorthogonal systems, and we study its relations with well -known cardinal functions. In particular, we obtain an original result relating the weight of a compact Hausdorff space K to the size of certain biorthogonal systems in C(K), generalizing a result of S.

In this research, the retro Banach frames are presented and different conditions are developed for the applications in the complex conjugate spaces. The retro Banach spaces are separable Banach spaces and used for the signal processing in the conjugate complex Banach spaces. Conventionally, the retro Banach spaces are applied to the Hilbert transform to be utilized in the conjugate Banach space. This research presents the further conditions for retro Banach frames that can be applied to the Wavelet and Gober’s transform such that this transform can be used in the conjugate complex Banach spaces. The current research discusses facts related to the form of Retro Banach Frames, Conjugate Banach Spaces, Linear Isomerism, Linear monomorphism, and Schauder basiss. It also helps in executing, extending, and modifying the Schauder frames to give a perfect characterization over the hesitation of Schauder frames.

Chapter 52 - The interplay between compact spaces and the Banach spaces of their continuous functions

Let X X be a separable Banach space. We give an almost characterization of the existence of a Finite Dimensional Decomposition (FDD for short) for X X in terms of Lipschitz retractions onto generating compact subsets K K of X X . In one direction, if X X admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set K K . On the other hand, we prove that if X X admits a “small” generating compact Lipschitz retract then X X has the π \pi -property. It is still unknown if the π \pi -property is isomorphically equivalent to the existence of an FDD. For dual Banach spaces this is true, so our results give a characterization of the FDD property for dual Banach spaces X X . We give an example of a small generating convex compact set which is not a Lipschitz retract of C [ 0 , 1 ] C[0,1] , although it is contained in a small convex Lipschitz retract and contains another one. We characterize isomorphically Hilbertian spaces as those Banach spaces X X for which every convex and compact subset is a Lipschitz retract of X X . Finally, we prove that a convex and compact set K K in any Banach space with a Uniformly Rotund in Every Direction norm is a uniform retract, of every bounded set containing it, via the nearest point map.

Let F be a Banach space. We establish necessary and sufficient conditions for the Dunford integration operator, from the space of F-valued Dunford integrable functions to the bidual of F, to belong to a given operator ideal. We also show how this fact can be used to characterize important classes of Banach spaces, such as Banach spaces with the Banach-Saks property, separable Banach spaces not containing c0, Banach spaces not containing c0 or ℓ1 and Asplund spaces not containing c0.

It is shown that for the vector space $\mathbb{R^N}$ (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a $\sigma$-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a~recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each $\sigma$-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a~linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete $\sigma$-finite Borel measure is solved positively. The uniqueness problem for non-$\sigma$-finite semi-finite translation invariant Borel measures on a Banach space $X$ which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure $\mu_B^0$ on $X$, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of $\mu_B^0$ and whose values on non-degenerate rectangles coincide with their volumes.

Asymptotically isometric copies of c0 and l1 in certain Banach spaces

The following dichotomy is proved. Every Banach space either contains a subspace isomorphic to ℓ 1 \ell ^1 , or it has an infinite-dimensional closed subspace which is a quotient of a Hereditarily Indecomposable (H.I.) separable Banach space. In the particular case of L p ( λ ) , 1 > p > ∞ L^p(\lambda ),\ 1>p>\infty , it is shown that the space itself is a quotient of a H.I. space. The factorization of certain classes of operators, acting between Banach spaces, through H.I. spaces is also investigated. Among others it is shown that the identity operator I : L ∞ ( λ ) → L 1 ( λ ) I: L^{\infty }(\lambda )\to L^1(\lambda ) admits a factorization through a H.I. space. The same result holds for every strictly singular operator T : ℓ p → ℓ q , 1 > p , q > ∞ T: \ell ^p\to \ell ^q,\ 1>p,q>\infty . Interpolation methods and the geometric concept of thin convex sets together with the techniques concerning the construction of Hereditarily Indecomposable spaces are used to obtain the above mentioned results.

Many of the fundamental research problems in the geometry of normed linear spaces can be loosely phrased as: Given a Banach space X and a class of Banach spaces Y does X contain a subspace Y ∈ Y? As a Banach space X is determined by its unit ball B x ≡ { x ∈ X :‖ x ‖ ≤ 1 } the problem can be rephrased in terms of the geometry of convex sets: Can a given unit ball B x be sliced with a subspace to obtain a set in some given class of unit balls? A result of this type is the famous theorem of Dvoretzky (see also [L], [M6], [M4], [MS], [FLM]).

Lipschitz Quotients from Metric Trees and from Banach Spaces Containing ℓ 1