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On some new characterizations of weakly compact sets in Banach spaces

Lixin Cheng, Qingjin Cheng, Zhenghua Luo (2010)

Studia Mathematica

We show several characterizations of weakly compact sets in Banach spaces. Given a bounded closed convex set C of a Banach space X, the following statements are equivalent: (i) C is weakly compact; (ii) C can be affinely uniformly embedded into a reflexive Banach space; (iii) there exists an equivalent norm on X which has the w2R-property on C; (iv) there is a continuous and w*-lower semicontinuous seminorm p on the dual X* with p ≥ s u p C such that p² is everywhere Fréchet differentiable in X*; and as a...

On some properties for dual spaces of Musielak-Orlicz function spaces

Zenon Zbąszyniak (2011)

Banach Center Publications

We will present relationships between the modular ρ* and the norm in the dual spaces ( L Φ ) * in the case when a Musielak-Orlicz space L Φ is equipped with the Orlicz norm. Moreover, criteria for extreme points of the unit sphere of the dual space ( L ⁰ Φ ) * will be presented.

On some properties of generalized Marcinkiewicz spaces

Evgeniy Pustylnik (2001)

Studia Mathematica

We give a full solution of the following problems concerning the spaces M φ ( X ⃗ ) : (i) to what extent two functions φ and ψ should be different in order to ensure that M φ ( X ⃗ ) ≠ M ψ ( X ⃗ ) for any nontrivial Banach couple X⃗; (ii) when an embedding M φ ( X ⃗ ) ⊊ M ψ ( X ⃗ ) can (or cannot) be dense; (iii) which Banach space can be regarded as an M φ ( X ⃗ ) -space for some (unknown beforehand) Banach couple X⃗.

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function f : [ 0 , 1 ] m → X is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function f : [ 0 , 1 ] m → X with respect to any norming subset there exists a separately increasing function g : [ 0 , 1 ] m → ℝ such that the sets of points of discontinuity...

On some properties of the class of stationary sets

Pascal Lefevre (1998)

Colloquium Mathematicae

Some new properties of the stationary sets (defined by G. Pisier in [12]) are studied. Some arithmetical conditions are given, leading to the non-stationarity of the prime numbers. It is shown that any stationary set is a set of continuity. Some examples of "large" stationary sets are given, which are not sets of uniform convergence.

On some properties of the functions from Sobolev-Morrey type spaces

Alik Najafov (2005)

Open Mathematics

In this paper the spaces of type Sobolev-Morrey-W p,a,г,τl(Q,G)-are constructed, the differential properties are studied and it is proved that the functions from these spaces satisfy Holder's condition, in the case, if the domain G∋R n satisfies the flexible λ-horn condition.

Currently displaying 941 – 960 of 1948