CONTENTS Introduction.................................................................................................................................................................5 Conventions and notations.........................................................................................................................................8 1. Preliminaries from the theory of locally convex spaces.........................................................................................11 2. Ultraproducts of locally convex spaces.................................................................................................................16 3. The principle of local reflexivity for locally convex spaces.....................................................................................19 4. Complemented subspaces of products and direct sums of Banach $L_p$-spaces and L₁ -predual spaces........24 5. Definition, examples and basic properties of locally convex $ℒ_p$-spaces..........................................................29 6. Definition, examples and basic properties of $Dℒ_p$-spaces..............................................................................39 7. Local properties of complemented subspaces of products and direct sums of Banach $L_p$-spaces................42 8. Duality between $ℒ_p$ and $Dℒ_p$-spaces and a characterization of injective spaces.....................................46 9. Extension, lifting and factorization properties of $ℒ_p$ and $Dℒ_p$-spaces.......................................................56 10. Fréchet $ℒ_p$- and (DF)-&Dℒ_p$-spaces with applications to Fréchet injective spaces..................................62 11. Comments and open problems...........................................................................................................................71 References...............................................................................................................................................................73
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This paper is an extended version of an invited talk presented during the Orlicz Centenary Conference (Poznań, 2003). It contains a brief survey of applications to classical problems of analysis of the theory of the so-called PLS-spaces (in particular, spaces of distributions and real analytic functions). Sequential representations of the spaces and the theory of the functor Proj¹ are applied to questions like solvability of linear partial differential equations, existence of a solution depending linearly and continuously on the right hand side of the equation and existence of a solution depending analytically on parameters.
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We give an elementary approach which allows us to evaluate Seip's conditions characterizing interpolating and sampling sequences in weighted Bergman spaces of infinite order for a wide class of weights depending on the distance to the boundary of the domain. Our results also give some information on cases not covered by Seip's theory. Moreover, we obtain new criteria for weights to be essential.
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Let Ω be an open connected subset of $ℝ^d$. We show that the space A(Ω) of real-analytic functions on Ω has no (Schauder) basis. One of the crucial steps is to show that all metrizable complemented subspaces of A(Ω) are finite-dimensional.
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We consider the topological algebra of (Taylor) multipliers on spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We describe multiplicative functionals and algebra homomorphisms on that algebra as well as idempotents in it. We show that it is never a Q-algebra and never locally m-convex. In particular, we show that Taylor multiplier sequences cease to be so after most permutations.
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We study those Köthe coechelon sequence spaces $k_{p}(V)$, 1 ≤ p ≤ ∞ or p = 0, which are locally convex (Riesz) algebras for pointwise multiplication. We characterize in terms of the matrix V = (vₙ)ₙ when an algebra $k_{p}(V)$ is unital, locally m-convex, a 𝒬-algebra, has a continuous (quasi)-inverse, all entire functions act on it or some transcendental entire functions act on it. It is proved that all multiplicative functionals are continuous and a precise description of all regular and all degenerate maximal ideals is given even for arbitrary solid algebras of sequences with pointwise multiplication. In particular, it is shown that all regular maximal ideals are solid.
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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.
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