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On the geometry of proportional quotients of l m

Piotr Mankiewicz, Stanisław J. Szarek (2003)

Studia Mathematica

We compare various constructions of random proportional quotients of l m (i.e., with the dimension of the quotient roughly equal to a fixed proportion of m as m → ∞) and show that several of those constructions are equivalent. As a consequence of our approach we conclude that the most natural “geometric” models possess a number of asymptotically extremal properties, some of which were hitherto not known for any model.

On the global stable manifold

Alberto Abbondandolo, Pietro Majer (2006)

Studia Mathematica

We give an alternative proof of the stable manifold theorem as an application of the (right and left) inverse mapping theorem on a space of sequences. We investigate the diffeomorphism class of the global stable manifold, a problem which in the general Banach setting gives rise to subtle questions about the possibility of extending germs of diffeomorphisms.

On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems

Robert E. Zink (2002)

Colloquium Mathematicae

In one of the earliest monographs that involve the notion of a Schauder basis, Franklin showed that the Gram-Schmidt orthonormalization of a certain Schauder basis for the Banach space of functions continuous on [0,1] is again a Schauder basis for that space. Subsequently, Ciesielski observed that the Gram-Schmidt orthonormalization of any Schauder system is a Schauder basis not only for C[0,1], but also for each of the spaces L p [ 0 , 1 ] , 1 ≤ p < ∞. Although perhaps not probable, the latter result would...

On the H -property of some Banach sequence spaces

Suthep Suantai (2003)

Archivum Mathematicum

In this paper we define a generalized Cesàro sequence space ces ( p ) and consider it equipped with the Luxemburg norm under which it is a Banach space, and we show that the space ces ( p ) posses property (H) and property (G), and it is rotund, where p = ( p k ) is a bounded sequence of positive real numbers with p k > 1 for all k N .

On the H-property and rotundity of Cesàro direct sums of Banach spaces

Saard Youyen, Suthep Suantai (2008)

Banach Center Publications

In this paper, we define the direct sum ( i = 1 n X i ) c e s p of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ( i = 1 n X i ) c e s p has the H-property if and only if each X i has the H-property, and ( i = 1 n X i ) c e s p has the Schur property if and only if each X i has the Schur property. Moreover, we also show that ( i = 1 n X i ) c e s p is rotund if and only if each X i is rotund.

On the isomorphism classes of weighted spaces of harmonic and holomorphic functions

Wolfgang Lusky (2006)

Studia Mathematica

Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of H v = f : Ω h o l o m o r p h i c : s u p z Ω | f ( z ) | v ( z ) < and investigate some isomorphism classes of h v = f : Ω h a r m o n i c : s u p z Ω | f ( z ) | v ( z ) < where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either H v l or H v H , and at least two possibilities for hv, again h v l and h v H . We also discuss many new examples of weights.

On the k-convexity of the Besicovitch-Orlicz space of almost periodic functions with the Orlicz norm

Fazia Bedouhene, Mohamed Morsli (2007)

Colloquium Mathematicae

Boulahia and the present authors introduced the Orlicz norm in the class B ϕ -a.p. of Besicovitch-Orlicz almost periodic functions and gave several formulas for it; they also characterized the reflexivity of this space [Comment. Math. Univ. Carolin. 43 (2002)]. In the present paper, we consider the problem of k-convexity of B ϕ -a.p. with respect to the Orlicz norm; we give necessary and sufficient conditions in terms of strict convexity and reflexivity.

On the KR and WKR points of Orlicz.

Yunan Cui, Yunfeng Zhang (1997)

Collectanea Mathematica

In this paper, we obtain criteria for KR and WKR points in Orlicz function spaces equipped with the Luxemburg norm.

On the lambda-property and computation of the lambda-function of some normed spaces.

Mohamed Akkouchi, Hassan Sadiky (1993)

Extracta Mathematicae

R. M. Aron and R. H. Lohman introduced, in [1], the notion of lambda-property in a normed space and calculated the lambda-function for some classical normed spaces. In this paper we give some more general remarks on this lambda-property and compute the lambda-function of other normed spaces, namely: B(S,∑,X) and Md(E).

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