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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 set Γ, (4) if...

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 points of discontinuity...

We say that an infinite, zero dimensional, compact Hausdorff space K has property (*) if for every nonempty open subset U of K there exists an open and closed subset V of U which is homeomorphic to K. We show that if K is a compact Hausdorff space with property (*) and X is a Banach space which contains a subspace isomorphic to the space C(K) of all scalar (real or complex) continuous functions on K and Y is a closed linear subspace of X which does not contain any subspace isomorphic to the space...

We show that the Banach space $D(0,1)$ of all scalar (real or complex) functions on $[0,1)$ that are right continuous at each point of $[0,1)$ with left-hand limit at each point of $(0,1]$ equipped with the uniform convergence topology is primary.

A Banach space $X$ contains an isomorphic copy of $C\left(\right[0,1\left]\right)$, if it contains a binary tree $\left(e_n\right)$ with the following properties (1) $e_n=e_{2n}+e_{2n+1}$ and (2) $c{max}_{2^n\le k{2}^{n+1}}|a_k|\le \parallel {\sum }_{k=2^n}^{2^{n+1}-1}{a}_k{e}_k\le C{max}_{2^n\le k{2}^{n+1}}\left|a_k\right|$ for some constants $0c\le C$ and every $n$ and any scalars $a_{2^n},\cdots ,a_{2^{n+1}-1}$. We present a proof of the following generalization of a Rosenthal result: if $E$ is a closed subspace of a separable $C\left(K\right)$ space with separable annihilator and$S:E\to X$ is a continuous linear operator such that $S^{\ast }$ has nonseparable range, then there exists a subspace $Y$ of $E$ isomorphic to $C\left(\right[0,1\left]\right)$ such that ${S|}_Y$ is an isomorphism, based on the fact....

Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes...

Let $J$ be an infinite set. Let $X$ be a real or complex $\sigma$-order continuous rearrangement invariant quasi-Banach function space over $({\{0,1\}}^J,{ℬ}^J,{\lambda }_J)$, the product of $J$ copies of the measure space $(\{0,1\},2^{0,1},\frac{1}{2}{\delta }_0+\frac{1}{2}{\delta }_1)$. We show that if $0p2$ and $X$ contains a function $f$ with the decreasing rearrangement $f^{\ast }$ such that $f^{\ast }\left(t\right)t^{-\frac{1}{p}}$ for every $t\in (0,1)$, then it contains an isometric copy of the Lebesgue space $L^p\left({\lambda }_J\right)$. Moreover, if $X$ contains a function $f$ such that $f^{\ast }\left(t\right)\sqrt{\left|\text{ln}\right(t\left)\right|}$ for every $t\in (0,1)$, then it contains an isometric copy of the Lebesgue space $L^2\left({\lambda }_J\right)$.

Let $S^p=\{S_t^p:t=\frac{k}{2^n},0\le k\le 2^n,n\in \mathbb{N}\}$ be a stochastic process on a probability space $(\Omega ,\Sigma ,P)$ with independent and time homogeneous increments such that $S_t^p-S_u^p$ is identically distributed as ${(t-u)}^{1/p}{Z}_p$ for each $0\le ut\le 1$ where $Z_p$ is a given symmetric $p$-stable distribution. We show that the closed linear hull of $S^p$ forms an isometric copy of the real Lebesgue space $L^p(0,1)$ in any quasi-Banach space $X$ consisting of $P$-a.e. equivalence classes of $\Sigma$-measurable real functions on $\Omega$ equipped with a rearrangement invariant quasi-norm which contains $S^p$ as a subset. It is possible...

AbstractThe paper contains studies of relationships between properties of the “translation” mappings $T_F$ and the topological and geometric structure of spaces X and Hardy classes $h^p(,X)$ of X-valued harmonic functions on the open unit disk in ℂ (X is a Banach space). The mapping $T_F$ transforming the unit circle of ℂ into $h^p(,X)$ is associated with a function $F\in h^p(,X)$ by the formula $T_F\left(t\right)=F\circ \varphi ₜ$, where ϕₜ is the rotation of through t.AcknowledgmentsThis work is based in part on the author’s doctoral thesis written at the Institute...