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.     

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     

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     

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.     

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).     

On the linear polarization constants of finite dimensional spaces

We study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spaces : they behave as if and as if . For we get the asymptotic behaviour up to a logarithmic factor.     

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.     

On bases,finite dimensional decompositions and weaker structures in Banach spaces

This is an investigation of the connections between bases and weaker structures in Banach spaces and their duals. It is proved, e.g., thatX has a basis ifX* does, and that ifX has a basis, thenX* has a basis provided thatX* is separable and satisfies Grothendieck’s approximation property; analogous results are obtained concerning π-structures and finite dimensional Schauder decompositions. The basic results are then applied to show that every separableℒ _p space has a basis. The second and third named authors have been supported by the NSF Grant GP 12997.     

Relations between rational and polynomial approximations in Banach spaces

В РАБОтЕ ИжУЧАУтсь Ус лОВИь, ДАУЩИЕ РАВЕНст ВА ИлИ стРОгИЕ НЕРАВЕНстВА МЕжДУ ВЕлИЧИНАМИ НАИ МЕНьшИх РАцИОНАльНы х И пОлИНОМИАльНых УклО НЕНИИ В БАНАхОВых пРОстРАНс тВАх. лль ОпРЕДЕлЕННО гО клАссА БАНАхОВых пРОстРАНс тВ НАИДЕНы ОгРАНИЧЕНИь НА ВОжМОжНОЕ кОлИЧЕс тВО НЕтРИВИАльНых сОВпА ДЕНИИ пОлИНОМИАльНы х И РАцИОНАльНых АппРО ксИМАцИИ РАВНых стЕп ЕНЕИ. РАссМОтРЕНы пРИБлИж ЕНИь пОДпРОстРАНстВАМИ Р АцИОНАльНых ФУНкцИИ с ФИксИРОВАННыМ “жНАМ ЕНАтЕлЕМ”. пОлУЧЕНы НЕОБхОДИМы Е УслОВИь Дль РАцИОНА льНОИ ФУНкцИИ НАИлУЧшЕгО пРИБлИжЕНИь. В жАклУЧ ИтЕльНОИ ЧАстИ ДОкАж АНА ВОжМОжНОсть сОВпАДЕ НИь НЕ РАВНых НУльУ ВЕлИЧИН НАИлУЧшИх пОлИНОМИА льНых стЕпЕНИn И РАцИОНАль Ных стЕпЕНИ (n, 1) пРИБлИжЕНИ И В пРОстРАНстВЕ хАРД ИH ₂(D).     

Groups acting on finite dimensional spaces with finite stabilizers

It is shown that every H -group G of type admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X ^H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a -module which is projective as a -module for all finite then M has finite projective dimension. Received: April 30, 1997     

Asymmetry and projection constants of Banach spaces

We discuss various asymmetry constants of finite-dimensional Banach spaces in a more generalized frame than that of [2], and solve a problem raised in [7] by finding an increasing sequence of Banach spaces whose diagonal asymmetry constants tend to infinity. We investigate the question of whether the projection constant of everyn-dimensional Banach space is strictly less than , and show that this is so whenn=2. The research for this paper was partially supported by NSF-GP-34193.     

Infinite dimensional Banach spaces of functions with nonlinear properties

The aim of this paper is to show that there exist infinite dimensional Banach spaces of functions that, except for 0, satisfy properties that apparently should be destroyed by the linear combination of two of them. Three of these spaces are: a Banach space of differentiable functions on ?ⁿ failing the Denjoy‐Clarkson property; a Banach space of non Riemann integrable bounded functions, but with antiderivative at each point of an interval; a Banach space of infinitely differentiable functions that vanish at infinity and are not the Fourier transform of any Lebesgue integrable function (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized degree for classes of finite dimensional upper hemi-continuous mappings in separable Banach spaces

Let E be a real separable Banach space, E^∗ the dual space of E, and Ω⊂E an open bounded subset, and let T:D(T)⊆E→2^E∗ be a finite dimensional upper hemi-continuous mapping with . A generalized degree theory is constructed for such a mapping. This degree is then applied to study the existence of approximate weak solutions to the equation 0∈Tx.     

On the inversion of some differentiable mappings with singularities between Banach spaces

Sunto. Il classico metodo di P. Levy e R. Caccioppoli è esteso al caso di applicazioni tra spazi di Banach che presentino singolarità. Viene fatta un'applicazione ad un problema non lineare in cui si ha modo di valutare esattamente il numero delle soluzioni.

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Entrata in Redazione il 12 novembre 1971.

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Supported by G.N.A.F.A. (Gruppo Nazionale di Analisi Funzionale ed Applicazioni, C.N.R.).     

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 .

     

Bicircular projections on some Banach spaces

In this paper we show that the bicircular projections are precisely the Hermitian projections and prove some immediate consequences of this result.