Inductive Limit Of Banach Spaces Research Articles

Mean Ergodic Multiplication Operators on Weighted Spaces of Continuous Functions

Multiplication operators on weighted Banach spaces and locally convex spaces of continuous functions have been thoroughly studied. In this note, we characterize when continuous multiplication operators on a weighted Banach space and on a weighted inductive limit of Banach spaces of continuous functions are power bounded, mean ergodic or uniformly mean ergodic. The behaviour of the operator on weighted inductive limits depends on the properties of the defining sequence of weights and it differs from the Banach space case.

On Algebras Which are Inductive Limits of Banach Spaces

We introduce algebras which are inductive limits of Banach spaces and carry inequalities which are counterparts of the inequality for the norm in a Banach algebra, and show that the algebra of holomorphic functions on a compact set is such an algebra. We then define an associated Wiener algebra, and prove the corresponding version of the well-known Wiener theorem. Finally, we consider factorization theory in these algebras, and in particular, in the associated Wiener algebra.

Operator Calculus of Differential Chains and Differential Forms

We describe a subspace of de Rham currents called “differential chains,” given as an inductive limit of Banach spaces. We make use of the viewpoint introduced by Mackey and developed by Whitney in which currents are treated as chains instead of cochains. This approach gives our space and its topology an intrinsic geometric definition. Moreover, our sequence of norms gives fine control over convergence and compactness, providing viable alternatives to the weak and flat topologies on currents. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in this setting, and we are also able to define a Cartesian wedge product. We conclude with an application of this framework, generalizing the Reynolds’ Transport Theorem to nonsmooth domains.

Dual characterization of absolutely representing systems in inductive limits of Banach spaces

We establish a dual criterion for absolutely representing systems in the spaces of a sufficiently broad class of inductive limits of sequences of Banach spaces. The result is stated without any additional restrictions on the system. We consider applications to the systems of partial fractions in the spaces of locally analytic functions.

Intersections of Fréchet Spaces and (LB)-Spaces

This article presents results about the class of locally convex spaces which are defined as the intersection E∩F of a Frechet space F and a countable inductive limit of Banach spaces E. This class appears naturally in analytic applications to linear partial differential operators. The intersection has two natural topologies, the intersection topology and an inductive limit topology. The first one is easier to describe and the second one has good locally convex properties. The coincidence of these topologies and its consequences for the spaces E ∩ F and E + F are investigated. 2000 Mathematics Subject Classification. Primary: 46A04, secondary: 46A08, 46A13, 46A45. The aim of this paper is to investigate spaces E ∩ F which are the intersection of a Frechet space F and an (LB)-space E. They appear in several parts of analysis whenever the space F is determined by countably many necessary (e.g. differentiability of integrability) conditions and E is the dual of such a space, in particular E is defined by a countable sequence of bounded sets which may also be determined by concrete estimates. Two natural topologies can be defined on E ∩ F : the intersection topology, which has seminorms easy to describe and which permits direct estimates, and a finer inductive limit topology which is defined in a natural way and which has good locally convex properties, e.g. E ∩F with this topology is a barrelled space. It is important to know when these two topologies coincide. It turns out that the locally convex properties of E ∩ F with the intersection topology are related to the completeness of the (LF)-space E + F . The present setting provides us with new interesting examples of (LF)-spaces. Recent progress on the study of (LF)-spaces, see [4, 14, 25, 27], is very important in our work. Our main results are Theorems 4 and 7. The Examples 5 and 10 show the main difficulties. Our article continues the research in [5], where the case when F is a Frechet Schwartz space (an (FS)-space) and E is the dual of an (FS)-space (a (DFS)-space) was analyzed. One of the original motivations for the research in [5] came up in the investigations of Langenbruch [18, 19, 20] about the surjectivity of linear partial differential operators P (D) with constant coefficients on spaces of ultradifferentiable functions and ultradistributions. The extension of {ω}-ultradifferentiable regularity for {ω}ultradistributions u such that P (D)u = 0 across a hypersurface has consequences for the surjectivity of P (D), in the same way that Holmgren’s uniqueness theorem is relevant in the discussions of surjectivity problems for P (D) on C∞(Ω), see [15, Chapters 8 and 10]. The extension of regularity for ultradistributions was studied by Langenbruch in [18]. In the classical case, treated by Hormander in [15, Section 11.3], one seeks conditions on a linear subspace V of RN and on The research of Jose Bonet was partially supported by FEDER and MCYT Proyecto no. BFM20012670.

An infinite-dimensional manifold structure for analytic Lie pseudogroups of infinite type

We construct an infinite-dimensional manifold structure adapted to analytic Lie pseudogroups of infinite type. More precisely, we prove that any isotropy subgroup of an analytic Lie pseudogroup of infinite type is a regular infinite-dimensional Lie group, modelled on a locally convex strict inductive limit of Banach spaces. This is an infinite-dimensional generalization to the case of Lie pseudogroups of the classical second fundamental theorem of Lie.

CONTINUOUS DEPENDENCE ON INITIAL DATA FOR SOLUTIONS OF NONLINEAR STOCHASTIC EVOLUTION EQUATIONS

We consider stochastic evolution equations in the framework of white noise analysis. Contraction operators on inductive limits of Banach spaces arise naturally in this context and we first extend Banach's fixed point theorem to this type of spaces. In order to apply the fixed point theorem to evolution equations, we construct a topological isomorphism between spaces of generalized random fields and the corresponding spaces of U-functionals. As an application we show that the solutions of some nonlinear stochastic heat equations depend continuously on their initial data. This method also applies to stochastic Volterra equations, stochastic reaction–diffusion equations and to anticipating stochastic differential equations.

On the parametrization problem of Lie pseudogroups of infinite type

We construct a local chart near the identity for analytic Lie pseudogroups of infinite type. The underlying manifold structure is modelled on an inductive limit of Banach spaces, endowed with the Silva differential calculus.

Inductive limits and geometry of Banach spaces

One of the main problems in the theory of inductive limits of Banach spaces is the projective description problem, finding a reasonable representation for the continuous seminorms. The problem is nontrivial even in the simplest cases. Recall that given, for example, an increasing sequence of Banach spaces (Yk)∞k=1 with continuous embeddings Yk[rarrhk ]Yk+1 the inductive limit is the space Y=∪kYk endowed with the finest locally convex topology τ such that every embedding Yk[rarrhk ](Y, τ) becomes continuous. It is possible to give abstract definitions for families of continuous seminorms generating the topology τ, but the connection with the norms of the step spaces Yk is not necessarily very close. For example, if the spaces Yk are Banach spaces of continuous functions endowed with weighted sup-norms, it is not clear if the continuous seminorms of the inductive limit are of the same type.We mention that inductive limits of spaces of continuous and holomorphic functions occur in many areas of analysis like linear partial differential operators, convolution equations [BD1], [E], complex and Fourier analysis and distribution theory. The projective description problem in these spaces has been thoroughly studied in [BMS1, BB1, BB2, BB3, BT, BM1, BM2], to mention some examples. We refer to the survey articles [BM1,BMS2, BB3]. The present work is also connected with the factorization problems which are treated in the book [Ju].

Hypercyclic Operators on Non-normable Fréchet Spaces

Every infinite dimensional separable non-normable Fréchet space admits a continuous hypercyclic operator. A large class of separable countable inductive limits of Banach spaces with the same property is given, but an example of a separable complete inductive limit of Banach spaces which admits no hypercyclic operator is provided. It is also proved that no compact operator on a locally convex space is hypercyclic.

Interpolation of Entire Functions and Projective Descriptions

Special entire functions of completely regular growth with additional properties are utilized to interpolate entire functions with certain bounds, and to give an example of a weighted inductive limit of Banach spaces of entire functions such that its topology cannot be described by the canonical weighted sup-seminorms associated with the weights of the steps. This provides a new, more natural counterexample to a problem of Bierstedt, Meise, and Summers.

Quasi‐incomplete regular LB‐space

A regular quasi-incomplete locally convex inductive limit of Banach spaces is constructed.

Holomorphic Functions on Strict Inductive Limits of Banach Spaces

In this article we show that a numberof apparently different properties coincide on tite set of holomorpitic functions on a strict inductive limit (aH inductive limits are asgumed to be countableand proper) ofBanach spaces and that they are ?II satisfied only 'u tite trivial case of a strict inductive limit offinite dimensional spaces. Thus tite linear properties of a strict inductive Iimit of Banach spaces rarely transiate themselVCS into itolomorphic properties.

Generation of locally equi-bounded semigroups in inductive limit of banach spaces

Generation of locally equi-bounded semigroups in inductive limit of banach spaces

Example of a sequentially incomplete regular inductive limit of Banach spaces

A sequentially incomplete regular inductive limit of a sequence of Banach spaces is constructed.

Stability and closed graph theorems in classes of bornological spaces

In the general theory of locally convex spaces, the idea of inductive limit is pervasive, with quotient spaces and the less obvious notion of direct sum being among the instances. Bornological spaces provide another important example. As is well known (cf. [7]), a Hausdorff locally convex space E is bornological if, and only if, E is an inductive limit of normed vector spaces. Going even further in this direction, a complete Hausdorff bornological space is an inductive limit of Banach spaces.

Inductive Limits of Banach Spaces

AbstractIf X and Y are infinite-dimensional Banach spaces, then Y is the inductive limit of Banach spaces each isomorphic to X.

On inductive limits of banach spaces

On inductive limits of banach spaces

Projective and inductive limits of Banach spaces

Projective and inductive limits of Banach spaces

Extension of sequentially continuous functionals in inductive limits of Banach spaces

Extension of sequentially continuous functionals in inductive limits of Banach spaces