Book contents
- Frontmatter
- Contents
- Foreword
- List of Main Lectures
- List of Participants
- A note on H. Ishihara and W. Takahashi modulus of convexity
- A property of non-strongly regular operators
- The entropy of convex bodies with ‘few’ extreme points
- Spaces of vector valued analytic functions and applications
- Notes on approximation properties in separable Banach spaces
- Moduli of complex convexity
- Grothendieck type inequalities and weak Hilbert spaces
- A weak topology characterization of l1 (m)
- Singular integral operators: a martingale approach
- Remarks about the interpolation of Radon-Nikodym operators
- Symmetric sequences in finite-dimensional normed spaces
- Some topologies on the space of analytic self-maps of the unit disk
- Minimal and strongly minimal Orlicz sequence spaces
- Type and cotype in Musielak-Orlicz spaces
- On the complex Grothendieck constant in the n-dimensional case
- Pathological properties and dichotomies for random quotients of finite-dimensional Banach spaces
- A note on a low M*-estimate
- The p1/p in Pisier's factorization theorem
- Almost differentiablity of convex functions in Banach spaces and determination of measures by their values on balls
- When E and E[E] are isomorphic
- A note on Gaussian measure of translates of balls
- Sublattices of M(X) isometric to M[0,1]
A weak topology characterization of l1 (m)
Published online by Cambridge University Press: 03 May 2010
- Frontmatter
- Contents
- Foreword
- List of Main Lectures
- List of Participants
- A note on H. Ishihara and W. Takahashi modulus of convexity
- A property of non-strongly regular operators
- The entropy of convex bodies with ‘few’ extreme points
- Spaces of vector valued analytic functions and applications
- Notes on approximation properties in separable Banach spaces
- Moduli of complex convexity
- Grothendieck type inequalities and weak Hilbert spaces
- A weak topology characterization of l1 (m)
- Singular integral operators: a martingale approach
- Remarks about the interpolation of Radon-Nikodym operators
- Symmetric sequences in finite-dimensional normed spaces
- Some topologies on the space of analytic self-maps of the unit disk
- Minimal and strongly minimal Orlicz sequence spaces
- Type and cotype in Musielak-Orlicz spaces
- On the complex Grothendieck constant in the n-dimensional case
- Pathological properties and dichotomies for random quotients of finite-dimensional Banach spaces
- A note on a low M*-estimate
- The p1/p in Pisier's factorization theorem
- Almost differentiablity of convex functions in Banach spaces and determination of measures by their values on balls
- When E and E[E] are isomorphic
- A note on Gaussian measure of translates of balls
- Sublattices of M(X) isometric to M[0,1]
Summary
Abstract. Let m be any cardinal. The main result characterizes ℓ1 (m) as the only Banach lattice whose positive cone is metrizable in the weak topology. Two related theorems on ℓ1-sums of Banach spaces are also proved.
Introduction
The three theorems of the paper concern ℓ1-sums of Banach spaces. The first, which states that the Banach space ℓ∞(m) contains the ℓ1-sum of 2m copies of itself, is essentially a translation into Banach space terms of a theorem of Pondiczery on the product topology [10]. The case m = ℵ0 of this result is reminiscent of the general theorem [9] that if X is any separable Banach space containing ℓ1, then X* contains M/(0, 1), the space of finite Borel measures on [0, 1]: the connection resides in the observation that M[0, l] is linearly isomorphic to the ℓ1-sum of c copies of L1 (0, 1). This observation is used to prove Theorem 2, which says that if X is as above then X* contains 2C mutually non-isomorphic closed linear subspaces.
Recall that the unit ball of a Banach space X is metrizable in the weak topology ifX* is separable, and recall too the consequence of the Baire category theorem that if X is infinite-dimensional then the weak topology on X and the weak-star topology on X* are not metrizable. However, there are still some interesting unbounded subsets of Banach spaces that are metrizable in the weak or the weak-star topology. Recall, for example, that the positive cone of the dual of C(K) (that is, the space of continuous functions on a compact metric space K), consisting of the non-negative finite Borel measures on K, is metrizable in the weak-star topology.
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- Geometry of Banach SpacesProceedings of the Conference Held in Strobl, Austria 1989, pp. 89 - 94Publisher: Cambridge University PressPrint publication year: 1991
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