### $\mathcal{U}\left(\mathcal{F}\right)$ algebras and the symbolic calculus of Sebastião e Silva. (Les algèbres $\mathcal{U}\left(\mathcal{F}\right)$ et le calcul symbolique de Sebastião e Silva.)

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We characterize when weighted $\left(LB\right)$-spaces of holomorphic functions have the dual density condition, when the weights are radial and grow logarithmically.

We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator $L:\u2107\left(K\right)\to {C}^{\infty}[0,1]$. At the same time, Markov’s inequality is not satisfied for certain polynomials on K.

We prove that for each countably infinite, regular space X such that ${C}_{p}\left(X\right)$ is a ${Z}_{\sigma}$-space, the topology of ${C}_{p}\left(X\right)$ is determined by the class ${F}_{0}\left({C}_{p}\left(X\right)\right)$ of spaces embeddable onto closed subsets of ${C}_{p}\left(X\right)$. We show that ${C}_{p}\left(X\right)$, whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set ${\Omega}_{\alpha}$ for the multiplicative Borel class ${M}_{\alpha}$ if ${F}_{0}\left({C}_{p}\left(X\right)\right)={M}_{\alpha}$. For each ordinal α ≥ 2, we provide an example ${X}_{\alpha}$ such that ${C}_{p}\left({X}_{\alpha}\right)$ is homeomorphic to ${\Omega}_{\alpha}$.

We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions ${c}_{p}\left(X\right)$ onto ${c}_{p}\left(X\right)$ × ℝ$.Inparticular,$cp(X)$isnotlinearlyhomeomorphicto$cp(X)$\times \mathbb{R}$. One of these examples is compact. This answers some questions of Arkhangel’skiĭ.

We show that Whitney?s approximation theorem holds in a general setting including spaces of (ultra)differentiable functions and ultradistributions. This is used to obtain real analytic modifications for differentiable functions including optimal estimates. Finally, a surjectivity criterion for continuous linear operators between Fréchet sheaves is deduced, which can be applied to the boundary value problem for holomorphic functions and to convolution operators in spaces of ultradifferentiable functions...

Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $$\overline{C}$$ (X) of C(X) such that the pair ($$\overline{C}$$ (X), C(X)) is homeomorphic to (Q, s). In case...

We characterize composition operators on spaces of real analytic functions which are open onto their images. We give an example of a semiproper map φ such that the associated composition operator is not open onto its image.

Sia $K\subset \subset {\mathbb{R}}^{n}$ un compatto, $f\ge 0$ una funzione analitica all'intorno di $K$, ed $m$ la massima molteplicità in $K$ degli zeri di $f$; si prova che la potenza ${f}^{\lambda}$ ($\lambda \in \u2102$, $Re\lambda >\mathrm{\u2014}\frac{1}{m}$) è integrabile in $K$. L'estensione meromorfa dell'applicazione $\lambda \to {f}^{\lambda}$ da $Re\lambda >0$ a tutto $\u2102$ (con valori in ${\mathcal{D}}^{\prime}(K)$ anziché in ${L}^{1}(K)$) era già stata provata in [1] e [2].