Let $X$ be a Banach space and $B\left(R\right)\subset X$ the ball of radius $R$ centered at $0$. Can any holomorphic function on $B\left(R\right)$ be approximated by entire functions, uniformly on smaller balls $B\left(r\right)$? We answer this question in the affirmative for a large class of Banach spaces.

In this article we examine necessary and sufficient conditions for the predual of the space of holomorphic mappings of bounded type, Gb(U), to have the approximation property and the compact approximation property and we consider when the predual of the space of holomorphic mappings, G(U), has the compact approximation property. We obtain also similar results for the preduals of spaces of m-homogeneous polynomials, Q(mE).

2000 Mathematics Subject Classification: 46B03We prove that any Lipschitz mapping from a separable Banach space into any Banach space can be approximated by uniformly Gâteaux differentiable Lipschitz mapping.Supported by grants GAUK 277/2001, GA CR 201-01-1198, AV 101-90-03. This paper is a part of PhD thesis prepared under the supervision of Professor Petr Hájek.

Soient $W=L^{\text{'}}\left(\mu \right)$ et $V=C\left(X\right)$. Il existe une application (non linéaire) normiquement continue $T\mapsto P\left(T\right)$ de l’espace des opérateurs bornés de $W$ dans $V$ sur l’espace des opérateurs compacts (resp. faiblement compacts) de $W$ dans $V$ telle que $\parallel T-P\left(T\right)\parallel$ coïncide avec la distance de $T$ au sous-espace formé des opérateurs compacts (resp. faiblement compacts). Pour un opérateur donné $T$ de $W$ dans $V$ on étudie les propriétés de l’ensemble $\mathbf{K}\left(T\right)$ (resp. $\mathbf{F}\left(T\right)$) des opérateurs compacts (resp. faiblement compacts) tel que pour tout $\mathbf{R}$ de $\mathbf{K}\left(T\right)$ (resp. $\mathbf{K}\left(T\right)$) la quantité...

We derive various approximation results in the theory of Hardy spaces on circular domains G. Two applications are given, one to operators which admit a nice representation of $H^{\infty }\left(G\right)$, and the other to extremal problems with links to the theory of differential equations.

Let X denote the space of all real, bounded double sequences, and let Φ, φ, Γ be φ-functions. Moreover, let Ψ be an increasing, continuous function for u ≥ 0 such that Ψ(0) = 0.In this paper we consider some spaces of double sequences provided with two-modular structure given by generalized variations and the translation operator (...).

Given an operator ideal ℐ, a Banach space E has the ℐ-approximation property if the identity operator on E can be uniformly approximated on compact subsets of E by operators belonging to ℐ. In this paper the ℐ-approximation property is studied in projective tensor products, spaces of linear functionals, spaces of linear operators/homogeneous polynomials, spaces of holomorphic functions and their preduals.

We obtain modular convergence theorems in modular spaces for nets of operators of the form $\left(T_{w}f\right)\left(s\right)={\int }_H{K}_w(s-h_w\left(t\right),f\left(h_w\left(t\right)\right))d{\mu }_H\left(t\right)$, w > 0, s ∈ G, where G and H are topological groups and ${h_w}_{w>0}$ is a family of homeomorphisms $h_w:H\to h_w\left(H\right)\subset G.$ Such operators contain, in particular, a nonlinear version of the generalized sampling operators, which have many applications in the theory of signal processing.