In this paper, we study stability of $M$-compactness for $l^p$ sum of Banach spaces for $1\leq p<\infty$. We also obtain a characterization of $M$-compact sets in terms of statistically maximizing sequence, a notion which is weaker than a maximizing sequence. Moreover, we introduce the notion of $\mathcal{I}$-$M$-compactness of a bounded subset $M$ of a normed linear space $X$ with respect to an ideal $\mathcal{I}$ and show that it is equivalent to $M$-compactness for non-trivial admissible ideals.
In this paper, we introduce and investigate a concept of Abel statistical continuity. A real valued function $f$ is Abel statistically continuous on a subset $E$ of $\R$, the set of real numbers, if it preserves Abel statistical convergent sequences, i.e. $(f(p_{k}))$ is Abel statistically convergent whenever $(p_{k})$ is an Abel statistical convergent sequence of points in $E$, where a sequence $(p_{k})$ of point in $\R$ is called Abel statistically convergent to a real number $L$ if Abel density of the set $\{k\in{\N}: |p_{k}-L|\geq\varepsilon \}$ is $0$ for every $\varepsilon>0$. Some other…
Considering the sets of subsums of series (or achievement sets) we show that for conditionally convergent series the multidimensional case is much more complicated than that of the real line. Although we are far from the full topological classification of such sets, we present many surprising examples and catch the ideas standing behind them in general theorems.
Levy-Steinitz theorem characterize sum range of conditionally convergent series, that is a set of all its convergent rearrangements; in finitely dimensional spaces -- it is an affine subspace. An achievement of a series is a set of all its subsums. We study the properties of achievement sets of series whose sum range is the whole plane. It turns out that it varies on the number of Levy vectors of a series.
In this paper, we generalized the Wijsman statistical convergence of closed sets in metric space by introducing the $f$-Wijsman statistical convergence these of sets, where $f$ is an unbounded modulus. It is shown that the Wijsman convergent sequences are precisely those sequences which are $f$-Wijsman statistically convergent for every unbounded modulus $f$. We also introduced a new concept of Wijsman strong Cesàro summability with respect to a modulus, and investigate the relationships between the $f$-Wijsman statistically convergent sequences and the Wijsman strongly Cesàro summable sequenc…
In this paper, we introduce a new class of subsets of bounded linear operators between Banach spaces which is p-version of the uniformly completely continuous sets. Then, we study the relationship between these sets with the equicompact sets. Moreover, we introduce the concept of weakly p-sequentially continuous differentiable mappings and obtain some characterizations of these mappings. Finally, we give a factorization result for differentiable mappings through p-convergent operators.
We study the concept of density for sets of natural numbers in some lacunary $A$-convergent sequence spaces. Also we are trying to investigate some relation between the ordinary convergence and module statistical convergence for evey unbounded modulus function. Morever we also study some results on the newly defined lacunary $f$-statistically $A$-convergent sequence spaces with respect to some Musielak-Orlicz function.
In this paper, we introduce the statistically multiplicative convergent sequences in locally solid Riesz algebras with respect to the algebra multiplication and the solid topology. We study on this concept and we give the notion of $\mathbb{st_m}$-bounded sequence, and also, we prove some relations between this convergence and the other convergences such as the order convergence and the statistical convergence in topological spaces. Also, we give some results related to semiprime $f$-algebras.
In this article, we introduce the notion of $p$-$(DPL)$ sets.\ Also, a factorization result for differentiable mappings through Dunford-Pettis $p$-convergent operators is investigated.\ Namely, if $ X ,Y $ are real Banach spaces and $U$ is an open convex subset of $X,$ then we obtain that, given a differentiable mapping $f: U\rightarrow Y$ its derivative $f^{\prime}$ takes $U$-bounded sets into $p$-$(DPL)$ sets if and only if it happens $f=g\circ S,$ where $S$ is a Dunford-Pettis $p$-convergent operator from $X$ into a suitable Banach space $Z$ and $g:S(U)\rightarrow Y$ is a Gâteaux differenti…
Let $\mathcal{I}\subseteq\mathcal{P}(ω)$ be a meager ideal. Then there are no continuous projections from $\ell_\infty$ onto the set of bounded sequences which are $\mathcal{I}$-convergent to $0$. In particular, it follows that the set of bounded sequences statistically convergent to $0$ is not isomorphic to $\ell_\infty$.
In this paper, we introduce some new $I_λ$-lacunary statistically convergent sequence spaces of order $α$ defined by a Musielak-Orlicz function. We study some relations between $I_λ$-lacunary statistically convergence with lacunary $I_λ$-summable sequences. Moreover we also study about the $I_λ$-lacunary statistically convergence of sequences in topological groups and give some important inclusion theorems.
A sequence $(x_n)$ in a locally solid Riesz space $(E,τ)$ is said to be statistically unbounded $τ$-convergent to $x\in E$ if, for every zero neighborhood $U$, $\frac{1}{n}\big\lvert\{k\leq n:\lvert x_k-x\rvert\wedge u\notin U\}\big\rvert\to 0$ as $n\to\infty$. In this paper, we introduce this concept and give the notions $st$-$u_τ$-closed subset, $st$-$u_τ$-Cauchy sequence, $st$-$u_τ$-continuous and $st$-$u_τ$-complete locally solid vector lattice. Also, we give some relations between the order convergence and the $st$-$u_τ$-convergence.
Leonetti proved that whenever $\mathcal I$ is an ideal on $\mathbb N$ such that there exists an~uncountable family of sets that are not in $\mathcal I$ with the property that the intersection of any two distinct members of that family is in $\mathcal I$, then the space $c_{0,\mathcal I}$ of sequences in $\ell_\infty$ that converge to 0 along $\mathcal I$ is not complemented. We provide a shorter proof of a more general fact that the quotient space $\ell_\infty / c_{0,\mathcal I}$ does not even embed into $\ell_\infty$.
An achievement set of a series is a set of all its subsums. We study the properties of achievement sets of conditionally convergent series in finite dimensional spaces. The purpose of the paper is to answer some of the open problems formulated in \cite{GM}. We obtain general results for series with harmonic-like coordinates, that is $A((-1)^{n+1}n^{-α_1},\dots,(-1)^{n+1}n^{-α_d})=\mathbb{R}^d$ for pairwise distinct numbers $α_1,\dots,α_d\in(0,1]$. For $d=2$, $α_1=1, α_2=\frac{1}{2}$ it was stated as an open problem in \cite{GM}, that is $A(\frac{(-1)^n}{n},\frac{(-1)^n}{\sqrt{n}})=\mathbb{R}^2…
The $p$-Gelfand Phillips property ($1\le p<\infty$) is studied in spaces of operators. Dunford - Pettis type like sets are studied in Banach spaces. We discuss Banach spaces $X$ with the property that every $p$-convergent operator $T:X\to Y$ is weakly compact, for every Banach space $Y$.
A Banach space $X$ is said to have property ($μ^s$) if every weak$^*$-null sequence in $X^*$ admits a subsequence such that all of its subsequences are Cesàro convergent to $0$ with respect to the Mackey topology. This is stronger than the so-called property (K) of Kwapień. We prove that property $(μ^s)$ holds for every subspace of a Banach space which is strongly generated by an operator with Banach-Saks adjoint (e.g. a strongly super weakly compactly generated space). The stability of property $(μ^s)$ under $\ell^p$-sums is discussed. For a family $\mathcal{A}$ of relatively weakly compact s…
A sequence $(x_n)$ in a lattice-normed space $(X,p,E)$ is statistical $p$-convergent to $x\in X$ if there exists a statistical $p$-decreasing sequence $q\stpd 0$ with an index set $K$ such that $δ(K)=1$ and $p(x_{n_k}-x)\leq q_{n_k}$ for every $n_k\in K$. This convergence has been investigated recently for $(X,p,E)=(E,|\cdot|,E)$ under the name of statistical order convergence and under the name of statistical multiplicative order convergence, and also, for taking $E$ as a locally solid Riesz space under the names statistically unbounded $τ$-convergence and statistically multiplicative converg…
We introduce the notion of ideally supported achievement sets for a series of real numbers. We analize their complexity and topological properties. We compare the notion of ideal achievement sets with the notion of ideally supported sum range of real series, considered by Filipów and Szuca. We complete Filipów and Szuca characterization of ideal sum ranges, [R. Filipów, P. Szuca, Rearrangement of conditionally convergent series on a small set, J. Math. Anal. Appl. 362 (2010), no. 1, 64-71.], and we obtain some generalization of Riemann's Theorem.
Let $\sum_{n=1}^\infty x_n$ be a conditionally convergent series in a Banach space and let $τ$ be a permutation of natural numbers. We study the set $\operatorname{LIM}(\sum_{n=1}^\infty x_{τ(n)})$ of all limit points of a sequence $(\sum_{n=1}^p x_{τ(n)})_{p=1}^\infty$ of partial sums of a rearranged series $\sum_{n=1}^\infty x_{τ(n)}$. We give full characterization of limit sets in finite dimensional spaces. Namely, a limit set in $\mathbb{R}^m$ is either compact and connected or it is closed and all its connected components are unbounded. On the other hand each set of one of these types is …
We show that a real sequence $x$ is convergent if and only if there exist a regular matrix $A$ and an $F_{σδ}$-ideal $\mathcal{I}$ on $\mathbf{N}$ such that the set of subsequences $y$ of $x$ for which $Ay$ is $\mathcal{I}$-convergent is of the second Baire category. This includes the cases where $\mathcal{I}$ is the ideal of asymptotic density zero sets, the ideal of Banach density zero sets, and the ideal of finite sets. The latter recovers an old result given by Keogh and Petersen in [J. London Math. Soc. \textbf{33} (1958), 121--123]. Our proofs are of a different nature and rely on recent…
It is well-known that every weakly convergent sequence in $\ell_1$ is convergent in the norm topology (Schur's lemma). Phillips' lemma asserts even more strongly that if a sequence $(μ_n)_{n\in\mathbb N}$ in $\ell_\infty'$ converges pointwise on $\{0,1\}^\mathbb N$ to $0$, then its $\ell_1$-projection converges in norm to $0$. In this note we show how the second category version of Schur's lemma, for which a short proof is included, can be used to replace in Phillips' lemma $\{0,1\}^\mathbb N$ by any of its subsets which contains all finite sets and having some kind of interpolation property f…
The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if and only if $X$ is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. {\bf (2)} Let $X$ be a non-reflexive Banach space. Then there exists a convex subset $A\subset X^*$ such that $A^{(1)}\neq {\bar{A}\,}^*$ (the latter denotes the weak$^*$ closure of $A$). {\bf (3)} Let $X$ be a quasi-reflexive Banach space and…
Suppose that $X$ is a Banach space. We will show that $X$ does not contain a copy of $c_0$ if and only if for each series which is not unconditionally convergent in $X$ respective sets coding all bounded subseries and rearrangements are meager. We use Bessaga-Pełczy{ń}ski $c_0-$Theorem and concept of uniformly unconditionally bounded series. Moreover we prove similar result for the ideal boundedness for a class of Baire ideals using Talagrand's characterisation.
Let $\mathcal{I},\mathcal{J}$ be two ideals on $\mathbf{N}$ which contain the family $\mathrm{Fin}$ of finite sets. We provide necessary and sufficient conditions on the entries of an infinite real matrix $A=(a_{n,k})$ which maps $\mathcal{I}$-convergent bounded sequences into $\mathcal{J}$-convergent bounded sequences and preserves the corresponding ideal limits. The well-known characterization of regular matrices due to Silverman--Toeplitz corresponds to the case $\mathcal{I}=\mathcal{J}=\mathrm{Fin}$. Lastly, we provide some applications to permutation and diagonal matrices, which extend se…
We present several characterizations of uo-convergent nets or sequences in spaces of continuous functions $C(Ω)$, $C_b(Ω)$, $C_0(Ω)$, and $C^\infty(Ω)$, extending results of [vdW18]. In particular, it is shown that a sequence uo-converges iff it converges pointwise on a co-meagre set. We also characterize order bounded sets in spaces of continuous functions. This leads to characterizations of order convergence.
