Related papers: On strongly separately continuous functions on seq…

We prove that for any $\alpha\in[0,\omega_1)$ there exists a strongly separately continuous function $f:\ell_\infty\to [0,1]$ such that $f$ belongs to the $(\alpha+1)$'th /$(\alpha+2)$'th/ Baire class and does not belong to the $\alpha$'th…

For a Tychonoff space $X$, we will denote by $USC_{p}(X)$ ($B_1(X)$) a set of all real-valued upper semicontinuous functions (a set of all Baire functions of class 1) defined on $X$ endowed with the pointwise convergence topology. In this…

We investigate strongly separately continuous functions on a product of topological spaces and prove that if $X$ is a countable product of real lines, then there exists a strongly separately continuous function $f:X\to\mathbb R$ which is…

Let CBV denote the Banach algebra of all continuous real-valued functions of bounded variation, defined in [0,1]. We show that the set of strongly singular functions in CBV is nonseparably spaceable. We also prove that certain families of…

A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel…

We present condition on higher order asymptotic behaviour of basic sequences in a Banach space ensuring the existence of bounded non-compact strictly singular operator on a subspace. We apply it in asymptotic $\ell_p$ spaces, $1\leq…

We prove the following two results. 1. If $X$ is a completely regular space such that for every topological space $Y$ each separately continuous function $f:X\times Y\to\mathbb R$ is of the first Baire class, then every Lindel\"of subspace…

It is solved the problem on construction of separately continuous functions on product of $n$ topological spaces with given restriction. In particular, it is shown that for every topological space $X$ and $n-1$ Baire class function $g:X\to…

Given an infinite matrix $M=(m_{nk})$ we study a family of sequence spaces $\ell_M^p$ associated with it. When equipped with a suitable norm $\|\cdot\|_{M,p}$ we prove some basic properties of the Banach spaces of sequences…

Let $U$ be an open set in $\mathbb{R}^d$. A continuous function $f\colon U \to \mathbb{R}$ is strongly nowhere differentiable if and only if for each $\gamma\in(0,1]$ and for each unit speed $C^{1,\gamma}$ curve $c\colon [a,b] \to U$, the…

We introduce and study certain type of variable exponent \ell^p spaces. These spaces will typically not be rearrangement-invariant but instead they enjoy a good local control of some geometric properties. We obtain some interesting examples…

Using complex methods combined with Baire's Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to…

In this note, we introduce a new class of separable Banach spaces, ${SD^p}[{\mathbb{R}^n}],\;1 \leqslant p \leqslant \infty$, which contain each $L^p$-space as a dense continuous and compact embedding. They also contain the nonabsolutely…

We study properties of strongly separately continuous mappings defined on subsets of products of topological spaces equipped with the topology of pointwise convergence. In particular, we give a necessary and sufficient condition for a…

We study Banach spaces X with a strongly asymptotic l_p basis (any disjointly supported finite set of vectors far enough out with respect to the basis behaves like l_p) which are minimal (X embeds into every infinite dimensional subspace).…

In this paper, we give necessary and sufficient conditions for the space B_1(X) of first Baire class functions on a Tychonoff space X, with pointwise topology, to be (strongly) sequentially separable.

It is obtained necessary and sufficient conditions of dependence on $\aleph$ coordinates for functions of several variables, each of which is a product of metrizable factors. The set of discontinuity points of such functions is…

We investigate the existence of well-ordered sequences of Baire 1 functions on separable metric spaces.

In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and…

It is proved that every function of finite Baire index on a separable metric space $K$ is a $D$-function, i.e., a difference of bounded semi-continuous functions on $K$. In fact it is a strong $D$-function, meaning it can be approximated…