Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^X$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_{\omega }$ is $F_{\sigma \delta }$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_{\omega }$. We show that $S_{\omega }$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology....

We study the boundary structure for w*-compact subsets of dual Banach spaces. To be more precise, for a Banach space X, 0 < ϵ < 1 and a subset T of the dual space X* such that ⋃ B(t,ϵ): t ∈ T contains a James boundary for $B_{X*}$ we study different kinds of conditions on T, besides T being countable, which ensure that $X*={\overline{spanT}}^{\left|\right|·\left|\right|}$. (SP) We analyze two different non-separable cases where the equality (SP) holds: (a) if $J:X\to 2^{B_{X*}}$ is the duality mapping and there exists a σ-fragmented map f: X → X* such that...

For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{\alpha }$, $Y_{\alpha }$ and $Z_{\alpha }$ such that (i) $f{X}_{\alpha },f{Y}_{\alpha },f{Z}_{\alpha }=\omega ₀$, where f is either trdef or ₀-trsur, (ii) $A\left(\alpha \right)-trind{X}_{\alpha }=\infty$ and $M\left(\alpha \right)-trind{X}_{\alpha }=-1$, (iii) $A\left(\alpha \right)-trind{Y}_{\alpha }=-1$ and $M\left(\alpha \right)-trind{Y}_{\alpha }=\infty$, and (iv) $A\left(\alpha \right)-trind{Z}_{\alpha }=M\left(\alpha \right)-trind{Z}_{\alpha }=\infty$ and $A(\alpha +1)\cap M(\alpha +1)-trind{Z}_{\alpha }=-1$. We also show that there exists no separable metrizable space $W_{\alpha }$ with $A\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$, $M\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$ and $A\left(\alpha \right)\cap M\left(\alpha \right)-trind{W}_{\alpha }=\infty$, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

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. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\overline{linf\left(\right[0,1\left]\right)}$ contains an order isomorphic copy of D(0,1), (2) $\overline{linf\left(Q\right)}$ contains an isomorphic copy of C([0,1]), (3) $\overline{linf\left(\right[0,1\left]\right)}/\overline{linf\left(Q\right)}$ contains an isomorphic copy of c₀(Γ) for some uncountable...

Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_{\sigma }$ set under that function is again $F_{\sigma }$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers...

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 $f:{[0,1]}^m\to X$ 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 $f:{[0,1]}^m\to X$ with respect to any norming subset there exists a separately increasing function $g:{[0,1]}^m\to ℝ$ such that the sets of...

There exists a universal control sequence ${p̅\left(m\right)}_{m=1}^{\infty }$ of increasing positive integers such that: Every infinite-dimensional separable Banach space X has a biorthogonal system xₙ,xₙ* with ||xₙ|| = 1 and ||xₙ*|| < K for each n such that, for each x ∈ X, $x={\sum }_{n=1}^{\infty }{x}_{\pi \left(n\right)}*\left(x\right)x_{\pi \left(n\right)}$ where π(n) is a permutation of n which depends on x but is uniformly controlled by ${p̅\left(m\right)}_{m=1}^{\infty }$, that is, $n_{n=1}^m\subseteq \pi {\left(n\right)}_{n=1}^{p̅\left(m\right)}\subseteq n_{n=1}^{p̅(m+1)}$ for each m.

A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_k}$ for which $su{p}_{f\in S_{X*}}limsu{p}_{k\to \infty }f\left(x_{n_k}\right)$ is attained at some f in the dual unit sphere $S_{X*}$. A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f\in S_{X*}$, there exists $g\in S_{X*}$ such that $limsu{p}_{n\to \infty }f\left(xₙ\right)<limin{f}_{n\to \infty }g\left(xₙ\right)$. Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

For a linear operator T in a Banach space let ${\sigma }_p\left(T\right)$ denote the point spectrum of T, let ${\sigma }_{p,n}\left(T\right)$ for finite n > 0 be the set of all $\lambda \in {\sigma }_p\left(T\right)$ such that dim ker(T - λ) = n and let ${\sigma }_{p,\infty }\left(T\right)$ be the set of all $\lambda \in {\sigma }_p\left(T\right)$ for which ker(T - λ) is infinite-dimensional. It is shown that ${\sigma }_p\left(T\right)$ is ${ℱ}_{\sigma }$, ${\sigma }_{p,\infty }\left(T\right)$ is ${ℱ}_{\sigma \delta }$ and for each finite n the set ${\sigma }_{p,n}\left(T\right)$ is the intersection of an ${ℱ}_{\sigma }$ set and a ${}_{\delta }$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an $F_{\sigma \delta \sigma }$-subset of X and contains a retract R so that $R\times E^{\omega }$ is not homeomorphic to $E^{\omega }$. This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^D$ the following four conditions are equivalent: (i) K is fragmented by $d_D$, where, for each S ⊂ D, $d_S(x,y)=sup\varrho \left(x\right(t),y(t\left)\right):t\in S$. (ii) For each countable subset...

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies ${\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\partial }_j{f}_j{\left(\epsilon \right)\left|\right|}^{p}d\mu \left(\epsilon \right){\le }^p{\int }_{-1,1ⁿ}{\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\delta }_j\Delta f_j\left(\epsilon \right){\left|\right|}^{p}d\mu \left(\epsilon \right)d\mu \left(\delta \right)$, where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${{\partial }_j}_{j=1}^n$ and $\Delta ={\sum }_{j=1}^n{\partial }_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by ${ⁿ}_p\left(X\right)$, we show that ${ⁿ}_p\left(X\right)\le {\sum }_{k=1}^{n}1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_i$ for which $\left(X_i\right)/ℐn\left(X_i\right)$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_i$ can be divided into subsets in such a way that if $X_i$ and $X_j$ are in different...

By Fin(X) (resp. $Fi{n}^k\left(X\right)$), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ₂(τ) be the Hilbert space with weight τ and $\ell {₂}^f\left(\tau \right)$ the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if $E=\ell {₂}^f\left(\tau \right)$ or E is an absorbing set in ℓ₂(τ) for one of the absolute Borel classes ${}_{\alpha }\left(\tau \right)$ and ${}_{\alpha }\left(\tau \right)$ of weight ≤ τ (α > 0) then Fin(E) and each $Fi{n}^k\left(E\right)$ are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X)...

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

We consider the class of compact spaces $K_A$ which are modifications of the well known double arrow space. The space $K_A$ is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of $K_A$ spaces and on the isomorphic classification of the Banach spaces $C\left(K_A\right)$.

In this paper, we define the direct sum ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the H-property if and only if each $X_i$ has the H-property, and ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the Schur property if and only if each $X_i$ has the Schur property. Moreover, we also show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ is rotund if and only if each $X_i$ is rotund.

Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ${\left({\alpha }_j\right)}_{j=1}^{\infty }$ of scalars, there exists a subsequence ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ such that either every subsequence of ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ defines a universal series, or no subsequence of ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ defines a universal series. In particular examples we decide which of the two cases holds.

Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f{\in }_m$ and f̂ be its Taylor series at $0\in {ℝ}^m$. Split the set ${\mathbb{N}}^m$ of exponents into two disjoint subsets A and B, ${\mathbb{N}}^m=A\cup B$, and decompose the formal series f̂ into the sum of two formal series G and H, supported by A and B, respectively. Do there exist $g,h{\in }_m$ with Taylor series at zero G and H, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some...

Plane $d$-webs have been studied a lot since their appearance at the turn of the 20th century. A rather recent and striking result for them is the theorem of Dufour, stating that the measurable conjugacies between 3-webs have to be analytic. Here, we show that even the set-theoretic conjugacies between two $d$-webs, $d\ge 3$ are analytic unless both webs are analytically parallelizable. Between two set-theoretically conjugate parallelizable $d$-webs, however, there always exists a nonmeasurable conjugacy;...

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space $K_{2n}$ such that $C\left(K_{2n}\right)$ has no uncountable (semi)biorthogonal sequence ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of at most 2n-1 points of $K_{2n}$, but has biorthogonal systems ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves...

Let $\Omega \subset {ℂ}^n$ be a bounded, simply connected $ℂ$-convex domain. Let $\alpha \in {\mathbb{Z}}_{+}^n$ and let $f$ be a function on $\Omega$ which is separately $C^{2{\alpha }_j-1}$-smooth with respect to $z_j$ (by which we mean jointly $C^{2{\alpha }_j-1}$-smooth with respect to $\mathrm{Re}{z}_j$, $\mathrm{Im}{z}_j$). If $f$ is $\alpha$-analytic on $\Omega \setminus f^{-1}\left(0\right)$, then $f$ is $\alpha$-analytic on $\Omega$. The result is well-known for the case ${\alpha }_i=1$, $1\le i\le n$, even when $f$ a priori is only known to be continuous.

Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power $X^{\omega }$ of any subspace X ⊂ Y is not universal for the class ₂ of absolute $G_{\delta \sigma }$-sets; moreover, if Y is an absolute $F_{\sigma \delta }$-set, then $X^{\omega }$ contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute $G_{\delta }$-set, then $X^{\omega }$ contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable...

Let $X$ be a Banach space of analytic functions on the open unit disk and $\Gamma$ a subset of linear isometries on $X$. Sufficient conditions are given for non-supercyclicity of $\Gamma$. In particular, we show that the semigroup of linear isometries on the spaces $S^p$ ($p>1$), the little Bloch space, and the group of surjective linear isometries on the big Bloch space are not supercyclic. Also, we observe that the groups of all surjective linear isometries on the Hardy space $H^p$ or the Bergman space $L_a^p$ ($1<p<\infty$, $p\ne 2$)...

We prove that if $X$ is a first countable space with property $\left(DC\left({\omega }_1\right)\right)$ and with a $G_{\delta }$-diagonal then the cardinality of $X$ is at most $𝔠$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$.

Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ${\left({\lambda }_j\right)}_{j=1}^{\infty }$ is a sequence of distinct positive numbers. Then $span1,x^{\lambda ₁},x^{\lambda ₂},...$ is dense in C[0,1] if and only if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)=\infty$. Moreover, if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)<\infty$, then every function from the C[0,1] closure of $span1,x^{\lambda ₁},x^{\lambda ₂},...$ can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an...

Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of ${\omega }^{\omega }$ there exists a continuous function $f:{\omega }^{\omega }\to X$ such that $f^{-1}\left(C₀\right)=D₀$ and $f^{-1}\left(C₁\right)=D₁$. We give several explicit examples of complete pairs of coanalytic sets.

Geometric structure of Cesàro function spaces $Ce{s}_p\left(I\right)$, where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that $Ce{s}_p[0,1]$ contains isomorphic and complemented copies of $l_q$-spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces $Ce{s}_p[0,1]$.

Let ${𝒫}_n$ denote the class of analytic functions $p\left(z\right)$ of the form $p\left(z\right)=1+c_n{z}^n+c_{n+1}{z}^{n+1}+\cdots$ in the open unit disc $𝕌$. Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for $p\left(z\right)$ concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.