This paper deals with function spaces of varying smoothness $B_p^{,s₀}\left(ℝⁿ\right)$, where the function :x ↦ s(x) determines the smoothness pointwise. Those spaces were defined in [2] and treated also in [3]. Here we prove results about interpolation, trace properties and present a characterization of these spaces based on differences.

We characterize the norm attaining bilinear forms on L1[0,1], and show that the set of norm attaining ones is not dense in the space of continuous bilinear forms on L1[0,1].

We present monotonicity theorems for index functions of N-fuctions, and obtain formulas for exact values of packing constants. In particular, we show that the Orlicz sequence space $l^{\left(N\right)}$ generated by the N-function N(v) = (1+|v|)ln(1+|v|) - |v| with Luxemburg norm has the Kottman constant $K\left(l^{\left(N\right)}\right)=N^{-1}\left(1\right)/N^{-1}(1/2)$, which answers M. M. Rao and Z. D. Ren’s [8] problem.

We discuss k-rotundity, weak k-rotundity, C-k-rotundity, weak C-k-rotundity, k-nearly uniform convexity, k-β property, C-I property, C-II property, C-III property and nearly uniform convexity both pointwise and global in Orlicz function spaces equipped with Luxemburg norm. Applications to continuity for the metric projection at a given point are given in Orlicz function spaces with Luxemburg norm.

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is the Choquet...

We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space c₀.

Nous étudions les sous-espaces biinvariants du shift usuel sur les espaces à poids$$L_{\omega }^2=\left\{f\in L^2\left(𝕋\right):\parallel f{\parallel }_{\omega }={\left(\sum _{n\in \mathbb{Z}}\left|f\left(n\right)\right|{\omega }^2\left(n\right)\right)}^{1/2}\<+\infty \right\},$$où $\omega \left(n\right)=(1+n{)}^p,n\ge 0$ et $\frac{\omega \left(n\right)}{(1+|n|{)}^p}\overrightarrow{n\to -\infty }+\infty$, pour un certain entier $p\ge 1$. Nous montrons que la trace analytique de tout sous-espace biinvariant est de type spectral, lorsque ${\sum }_{n\ge 2}\frac{1}{nlog\omega (-n)}$ diverge, mais que ceci n’est plus valable lorsque ${\sum }_{n\ge 2}\frac{1}{nlog\omega (-n)}$ converge.

Cet article précise la notion de privilège introduite par A. Douady. Un sous-espace privilégié d’un polycylindre $K$ est défini par un idéal fermé de l’algèbre des fonctions continues sur $K$ et holomorphes sur $\dot{K}$, cet idéal étant supposé de résolution finie.Les sous-espaces privilégiés d’un polycylindre fixé sont classés par un espace analytique banachique, “une grassmannienne”, introduit par A. Douady et dont on donne ici la propriété universelle.Pour cela on montre que la notion de privilège est locale...

Several equivalent conditions are given for the existence of real-valued Baire functions of all classes on a type of $\mathbf{K}$-analytic spaces, called disjoint analytic spaces, and on all pseudocompact spaces. The sequential stability index for the Banach space of bounded continuous real-valued functions on these spaces is shown to be either $0,1$, or $\Omega$ (the first uncountable ordinal). In contrast, the space of bounded real-valued Baire functions of class 1 is shown to contain closed linear subspaces with index...

The classical level function construction of Halperin and Lorentz is extended to Lebesgue spaces with general measures. The construction is also carried farther. In particular, the level function is considered as a monotone map on its natural domain, a superspace of $L^p$. These domains are shown to be Banach spaces which, although closely tied to $L^p$ spaces, are not reflexive. A related construction is given which characterizes their dual spaces.