Covering spheres of Banach spaces by balls

If the unit sphere of a Banach space X can be covered by countably many balls no one of which contains the origin, then, as an easy consequence of the separation theorem, X* is w*-separable. We prove the converse under suitable renorming. Moreover, the balls of the countable covering can be chosen as translates of the same ball. Research of V. P. Fonf was supported in part by Israel Science Foundation, Grant # 139/02 and by the Istituto Nazionale di Alta Matematica of Italy. Research of C. Zanco was supported in part by the Ministero dell’Università e della Ricerca Scientifica e Tecnologica of Italy and by the Center for Advanced Studies in Mathematics at the Ben-Gurion University of the Negev, Beer-Sheva, Israel.     

Volumetric bounds for intersections of congruent balls in Euclidean spaces

We investigate the intersections of balls of radius r, called r-ball bodies, in Euclidean d-space. An r-lense (resp., r-spindle) is the intersection of two balls of radius r (resp., balls of radius r containing a given pair of points). We prove that among r-ball bodies of a given volume, the r-lense (resp., r-spindle) has the smallest inradius (resp., largest circumradius). In general, we upper (resp., lower) bound the intrinsic volumes of r-ball bodies of a given inradius (resp., circumradius). This complements and extends some earlier results on volumetric estimates for r-ball bodies.

     

M-structure and intersection properties of balls in Banach spaces

This paper contains a complex version of Theorem 5.4 (Dominated Extension Theorem) from the paper “Structure in real Banach spaces” by E. Effros and the present author (to appear in Annals of Mathematics).     

Embeddings of locally finite metric spaces into Banach spaces

We show that if is a Banach space without cotype, then every locally finite metric space embeds metrically into .

     

Finitely locally finite coverings of Banach spaces

A well-known result due to H. Corson states that, for any covering τ by closed bounded convex subsets of any Banach space X containing an infinite-dimensional reflexive subspace, there exists a compact subset C of X that meets infinitely many members of τ. We strengthen this result proving that, even under the weaker assumption that X contains an infinite-dimensional separable dual space, an (algebraically) finite-dimensional compact set C with that property can always be found.     

On finite dimensional subspaces of Banach spaces

The common Banach spaces are investigated with respect to some properties of their finite dimensional subspaces. The contribution to the paper of the second author is part of his Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor A. Dvoretzky. The authors wish to thank Professor A. Dvoretzky for the interest he showed in the paper and for his helpful advice.     

Sets of constant width in finite dimensional Banach spaces

In Euclidean space a set of constant width has the property that it is not a proper subset of any set of the same diameter. The converse implication is also true. Here we show that if Euclidean is replaced byn-dimensional Banach space the direct statement is true, but the converse statement is false. Attention is drawn to the problem of characterising those Banach spaces of finite dimension for which the converse is true.     

On Banach spaces whose dual balls are not weak∗ sequentially compact

Theorem 1. LetX be a Banach space. (a) IfX ^∗ has a closed subspace in which no normalized sequence converges weak^∗ to zero, thenl ₁ is isomorphic to a subspace ofX. (b) IfX ^∗ contains a bounded sequence which has no weak^∗ convergent subsequence, thenX contains a separable subspace whose dual is not separable. The second-named author was supported in part by NSF-MPS 72-04634-A03.     

Almost locally minimal projections in finite dimensional Banach spaces

A projectionP on a Banach spaceX is called “almost locally minimal” if, for every α>0 small enough, the ballB(P,α) in the spaceL(X) of all operators onX contains no projectionQ with whereD is a constant. A necessary and sufficient condition forP to be almost locally minimal is proved in the case of finite dimensional spaces. This criterion is used to describe almost locally minimal projections on ℓ ₁ ⁿ . Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas, 1997. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Integrals and Banach spaces for finite order distributions

Let B _c denote the real-valued functions continuous on the extended real line and vanishing at −∞. Let B _r denote the functions that are left continuous, have a right limit at each point and vanish at −∞. Define A _c ⁿ to be the space of tempered distributions that are the nth distributional derivative of a unique function in B _c . Similarly with A _r ⁿ from B _r . A type of integral is defined on distributions in A _c ⁿ and A _r ⁿ . The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ, the spaces A _c ⁿ and A _r ⁿ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to B _c and B _r , respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space A _c ¹ is the completion of the L ¹ functions in the Alexiewicz norm. The space A _r ¹ contains all finite signed Borel measures. Many of the usual properties of integrals hold: H?lder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.     

Relatively weakly open sets in closed balls of Banach spaces,and the centralizer

Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.     

Averaging distances in real quasihypermetric Banach spaces of finite dimension

The average distance theorem of Gross implies that for each realN-dimensional Banach space (N≥2) there is a unique positive real numberr(E) with the following property: For each positive integern and for all (not necessarily distinct)x ₁,x ₂, …,x _n inE with ‖x ₁‖=‖x ₂‖=…=‖x _n‖=1, there exists anx inE with ‖x‖=1 such that The main result of this paper shows, thatr(E)≤2−1/N for each realN-dimensional Banach spaceE (N≥2) with the so-called quasihypermetric property (which is equivalent toE isL ¹-embeddable). Moreover, equality holds if and only ifE is isometrically isomorphic to ℝ ^N equipped with the usual 1-norm.     

Invariant subspaces of finite codimension in Banach spaces of analytic functions

We give a characterization of invariant subspaces of finite codimension in Banach spaces of vector-valued analytic functions in several variables, where invariant refers to invariance under multiplication by any polynomial. We obtain very weak conditions under which our characterization applies, that unifies and improves a number of previous results. In the vector-valued case, the results are new even for one complex variable. As a concrete application in several variables, we consider the Bergman space on a strictly pseudo-convex domain, and we improve previous results (assuming C^∞-boundary) to the case of C²-boundary.     

Simplices with edges of equal length in finite dimensional Banach spaces

Natural generalizations of planar results on volume ratios with respect to convex bodies in finite dimensional Banach spaces are given. The underlying paper contains inequalities for the volume of simplices with edges of equal length in a Banach space.Dedicated to Professor Helmut Mäurer on the occasion of his 60th birthday     

Banach ideals of operators with applications to the finite dimensional structure of Banach spaces

We present a few applications of the theory of Banach ideals of operators. In particular, we give operator characterizations of the ℒ _p spaces, compute the relative projection constant of isometric embeddings of Hilbert spaces inL _p -spaces, and show that Π₁ (E, F), the space of absolutely summing operators, is reflexive ifE andF are reflexive andE has the approximation property. Research supported by NSF-GP-34193 Research supported by NSF-Science Development Grant     

On the uniqueness of symmetric bases in finite dimensional Banach spaces

Suppose{e _i} _i=1 ⁿ and{f _i} _i=1 ⁿ are symmetric bases of the Banach spacesE andF. Letd(E,F)≦C andd(E,l _n ² )≧n' for somer>0. Then there is a constantC _r=C_r(C)>0 such that for alla _i∈Ri=1,...,n $$C_r^{ - 1} \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\| \leqq \left\| {\sum\limits_{i = 1}^n {a_i f_i } } \right\| \leqq C_r \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\|$$ We also give a partial uniqueness of unconditional bases under more restrictive conditions.