It is an open question when the direct sum of normed spaces inherits uniform rotundity in every direction from the factor spaces. M. Smith [4] showed that, in general, the answer is negative. The purpose of this paper is to carry out a complete study of Smith's counterexample.
In any separable Banach space containing c 0 which admits a C k-smooth bump, every continuous function can be approximated by a C k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set...
There is given a criterion for an arbitrary element from the unit sphere of Musielak-Orlicz function space equipped with the Luxemburg norm to be a point of smoothness. Next, as a corollary, a criterion of smoothness of these spaces is given.
We show that, if μ is a probability measure and X is a Banach space, then the space L¹(μ,X) of Bochner integrable functions admits an equivalent Gâteaux (or uniformly Gâteaux) smooth norm provided that X has such a norm, and that if X admits an equivalent Fréchet (resp. uniformly Fréchet) smooth norm, then L¹(μ,X) has an equivalent renorming whose restriction to every reflexive subspace is Fréchet (resp. uniformly Fréchet) smooth.
First, we extend the criteria for smooth points of from [22] to the whole class of Musielak-Orlicz spaces. Next, we present criteria for very smooth and strongly smooth points of .
This is a short survey on some recent as well as classical results and open problems in smoothness and renormings of Banach spaces. Applications in general topology and nonlinear analysis are considered. A few new results and new proofs are included. An effort has been made that a young researcher may enjoy going through it without any special pre-requisites and get a feeling about this area of Banach space theory. Many open problems of different level of difficulty are discussed. For the reader...
Sobczyk's theorem asserts that every c₀-valued operator defined on a separable Banach space can be extended to every separable superspace. This paper is devoted to obtaining the most general vector valued version of the theorem, extending and completing previous results of Rosenthal, Johnson-Oikhberg and Cabello. Our approach is homological and nonlinear, transforming the problem of extension of operators into the problem of approximating z-linear maps by linear maps.
Sobczyk's theorem is usually stated as: every copy of c0 inside a separable Banach space is complemented by a projection with norm at most 2. Nevertheless, our understanding is not complete until we also recall: and c0 is not complemented in l∞. Now the limits of the phenomenon are set: although c0 is complemented in separable superspaces, it is not necessarily complemented in a non-separable superspace, such as l∞.
Several order relations in the set of strict t-norms are investigated.