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The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation

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Brzdęk J.

Annales Polonici Mathematici

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1996

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tom 64

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nr 3

195-205

EN

Let K denote the set of all reals or complex numbers. Let X be a topological linear separable F-space over K. The following generalization of the result of C. G. Popa [16] is proved. Theorem. Let n be a positive integer. If a Christensen measurable function f: X → K satisfies the functional equation $f(x + f(x)^ny) = f(x)f(y)$, then it is continuous or the set {x ∈ X : f(x) ≠ 0} is a Christensen zero set.

2

Bayoumi quasi-differential is not different from Fréchet-differential

100%

Albiac F., Ansorena J.

Open Mathematics

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2012

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tom 10

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nr 3

1071-1075

EN

Unlike for Banach spaces, the differentiability of functions between infinite-dimensional nonlocally convex spaces has not yet been properly studied or understood. In a paper published in this Journal in 2006, Bayoumi claimed to have discovered a new notion of derivative that was more suitable for all F-spaces including the locally convex ones with a wider potential in analysis and applied mathematics than the Fréchet derivative. The aim of this short note is to dispel this misconception, since it could hinder making headway in this already hard enough subject. To that end we show that Bayoumi quasi-differentiability, when properly defined, is the same as Fréchet differentiability, and that some of his alleged applications are wrong.

3

Copies of the sequence space $\omega$ in $F$-lattices with applications to Musielak−Orlicz spaces

100%

Wójtowicz M., Wiśniewska H.

Commentationes Mathematicae

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2016

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tom 56

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nr 1

EN

Let $E$ be a fixed real function $F$-space, i.e., $E$ is an order ideal in $L_0(S,\Sigma,\mu)$ endowed with a monotone $F$-norm $\|\|$ under which $E$ is topologically complete. We prove that $E$ contains an isomorphic (topological) copy of $\omega$, the space of all sequences, if and only if $E$ contains a lattice-topological copy $W$ of $\omega$. If $E$ is additionally discrete, we obtain a much stronger result: $W$ can be a projection band; in particular, $E$ contains a~complemented copy of $\omega$. This solves partially the open problem set recently by W. Wnuk. The property of containing a copy of $\omega$ by a Musielak−Orlicz space is characterized as follows. (1) A sequence space $\ell_{\Phi}$, where $\Phi = (\varphi_n)$, contains a copy of $\omega$ iff $\inf_{n \in \mathbb{N}} \varphi_n (\infty) = 0$, where $\varphi_n (\infty) = \lim_{t \to \infty} \varphi_n (t)$. (2) If the measure $\mu$ is atomless, then $\omega$ embeds isomorphically into $L_{\mathcal{M}} (\mu)$ iff the function $\mathcal{M}_{\infty}$ is positive and bounded on some set $A\in \Sigma$ of positive and finite measure, where $\mathcal{M}_{\infty} (s) = \lim_{n \to \infty} \mathcal{M} (n, s)$, $s\in S$. In particular, (1)' $\ell_\varphi$ does not contain any copy of $\omega$, and (2)' $L_{\varphi} (\mu)$, with $\mu$ atomless, contains a~copy $W$ of $\omega$ iff $\varphi$ is bounded, and every such copy $W$ is uncomplemented in $L_{\varphi} (\mu)$.

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On vector measures which have everywhere infinite variation or noncompact range

53%

Drewnowski L., Lipecki Z.

Instytut Matematyczny Polskiej Akademii Nauk

EN

Abstract Let X be an infinite-dimensional Banach space, let Σ be a σ-algebra of subsets of a set S, and denote by ca(Σ,X) the Banach space of X-valued measures on Σ equipped with the uniform norm. We say that a nonzero μ ∈ ca(Σ,X) is everywhere of infinite variation [has everywhere noncompact range] if |μ|(A) = ∞ or 0 [{μ(E): E ∈ Σ, E ⊂ A} is not relatively compact or equals {0}] for every A ∈ Σ. Let λ be a nonatomic probability measure on Σ, and denote by ca(Σ,λ,X) the closed subspace of ca(Σ,X) consisting of λ-continuous measures. Analogously as above, we define measures in ca(Σ,λ,X) that are λ-everywhere of infinite variation or have λ-everywhere noncompact range. Using the Dvoretzky-Rogers theorem, we give two constructions of an absolutely convergent series of λ-simple measures in ca(Σ,λ,X) such that the sum of each of its subseries is λ-everywhere of infinite variation. In particular, the normed space P(λ,X) of Pettis λ-integrable functions with values in X lacks property (K), and so is incomplete. These results refine and improve some earlier results of E. Thomas, and L. Janicka and N. J. Kalton. One of the constructions also yields the existence of an infinite-dimensional closed subspace in ca(Σ,λ,X) all of whose nonzero members are λ-everywhere of infinite variation. Moreover, modifying some ideas of R. Anantharaman and K. M. Garg, we prove that the measures that are λ-everywhere of infinite variation form a dense $G_δ$-set in ca(Σ,λ,X). From this we derive an analogous result on measures that are everywhere of infinite variation and the closed subspace of ca(Σ,X) consisting of nonatomic measures. Similar results concerning measures that have [λ-] everywhere noncompact range are also established. In this case, the existence of X-valued measures with noncompact range must, however, be postulated. We also prove that the measures of σ-finite variation form an $F_{σδ}$-, but not $F_σ$-, subset of ca(Σ,λ,X), and the same is true for P(λ,X) provided that X is separable. Finally, we consider the special case when X is a Banach lattice and, for X nonisomorphic to an AL-space, we note analogues of some of the results above for positive X-valued measures on Σ.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Commentationes Mathematicae

1 Open Mathematics

1 Albiac F.

1 Ansorena J.

1 Brzdęk J.

1 Drewnowski L.

1 Lipecki Z.

1 Wiśniewska H.

1 Wójtowicz M.

1 2016

1 2012

1 1996

1 1995

The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation

Bayoumi quasi-differential is not different from Fréchet-differential

Copies of the sequence space \(\omega\) in \(F\)-lattices with applications to Musielak−Orlicz spaces

On vector measures which have everywhere infinite variation or noncompact range