An example of a nuclear space in infinite dimensional holomorphy

LetU be an open subset of a complex locally convex spaceE, andH(U) the space of holomorphic functions fromU toC. If the dualE′ ofE is nuclear with respect to the topology generated by the absolutely convex compact subsets ofE, then it is shown thatH(U) endowed with the compact open topology is a nuclear space. In particular, ifE is the strong dual of a Fréchet nuclear space, thenH(U) is a Fréchet nuclear space.     

The bounded approximation property for spaces of holomorphic mappings on infinite dimensional spaces

We study the bounded approximation property for spaces of holomorphic functions. We show that if U is a balanced open subset of a Fréchet–Schwartz space or (DFM )‐space E , then the space ??(U ) of holomorphic mappings on U , with the compact‐open topology, has the bounded approximation property if and only if E has the bounded approximation property. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Banach-Dieudonné theorem for germs of holomorphic functions

Let $\text{H}$(U) denote the space of all holomorphic functions on an open subset U of a complex Fréchet space E. Let $\text{H}$(K) denote the space of all holomorphic germs on a compact subset K of E. It is shown that $\text{H}$(K), with a natural topology, is the inductive limit of a suitable sequence of compact subsets, within the category of all topological spaces. As an application of this result it is shown that the compact-ported topology introduced by Nachbin coincides with the compact-open topology on $\text{H}$(U) whenever U is a balanced open subset of a Fréchet-Schwartz space. This last result improves earlier results of P. Boland and S. Dineen [Bull. Soc. Math. France106 (1978), 311–336], R. Meise [Proc. Roy. Irish Acad. Sect. A81 (1981), 217–223], and others.     

A quantitative version of Krein’s theorems for Fréchet spaces

For a Banach space E and its bidual space E ^′′, the following function ${k(H) : = {\rm sup}_{y\in\overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}} {\rm inf}_{x\in E} \|y - x\|}$ defined on bounded subsets H of E measures how far H is from being σ(E, E′)-relatively compact in E. This concept, introduced independently by Granero [10] and Cascales et al. [7], has been used to study a quantitative version of Krein’s theorem for Banach spaces E and spaces C _p (K) over compact K. In the present paper, a quantitative version of Krein’s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E the above function k(H) reads as follows ${k(H) := {\rm sup}\{d(h, E) : h \in \overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}\},}$ where d(h, E) is the natural distance of h to E in the bidual E ^′′. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds ${k(coH) < (2^{n+1} - 2) k(H) + \frac{1}{2^{n}}}$ for all ${n \in \mathbb{N}}$ . Consequently this yields also the following formula ${k(coH) \leq \sqrt{k(H)}(3 - 2\sqrt{k(H)})}$ . Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein’s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet space. We also define and discuss two other measures of weak non-compactness lk(H) and k′(H) for a Fréchet space and provide two quantitative versions of Krein’s theorem for both functions.     

On the distinguished character of the function spaces of holomorphic mappings of bounded type

Given two complex normed spaces E and F, F complete, and a balanced open subset U of E, we prove that the space $\text{H}$_(b(U, F) of the holomorphic mappings f: U → F of bounded type, endowed with its natural topology τ_b, is a distinguished quasi-normable Fréchet space, which is not a Schwartz space unless dim E < ∞ and dim F < ∞.     

ON THE PRODUCT OF GATEAUX DIFFERENTIABILITY LOCALLY CONVEX SPACES

A locally convex space is said to be a Gâteaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gâteaux differentiable in D. This paper shows that the product of a GDS and a family of separable Fréchet spaces is a GDS, and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.     

Modified construction of nuclear Fréchet spaces without basis

Recently, B. Mitiagin and N. Zobin constructed an example of nuclear Fréchet space without basis. The essential modification of their constructions gives the following results. There exists such a nuclear Fréchet space X that for any nuclear Fréchet space Y the space X × Y has no basis (Sections 1 and 2). This fact has a lot of corollaries (Sect. 3); e.g., the space X × C^∞(R¹) having the maximal diametral dimension among nuclear Fréchet spaces nevertheless has no basis. One can also construct (Sect. 4) a nuclear Fréchet space $\text{X}\text{?}$ without strongly finite-dimensional decomposition (see Definition 0.1). In Section 5 some comments and open questions are given.     

Convolution operators on spaces of entire functions

We show that nontrivial convolution operators on certain spaces of entire functions on E are frequently hypercyclic when E is a normed space and when E is the strong dual of a Fréchet nuclear space. We also obtain results of existence and approximation for convolution equations on certain spaces of entire functions on arbitrary locally convex spaces.     

Subdifferentials of a minimal time function in normed spaces

In a general normed vector space, we study the minimal time function determined by a differential inclusion where the set-valued mapping involved has constant values of a bounded closed convex set U and by a closed target set S. We show that proximal and Fréchet subdifferentials of a minimal time function are representable by virtue of corresponding normal cones of sublevel sets of the function and level or suplevel sets of the support function of U. The known results in the literature require the set U to have the origin as an interior point or U be compact. (In particular, if the set U is the unit closed ball, the results obtained reduce to the subdifferential of the distance function defined by S.)     

On pure uniform holomorphy in spaces of holomorphic germs

If E is a complex (DFC)-space (see § 2), we show that E leads to pure uniform holomorphy (see §2) if and only if its Fréchet dual space E′ is separable (see Theorem 1, where these two conditions have other eight equivalent ones). By using a theorem of Mujica (see §4), we consider the (DFC)-space K(K) of germs around K of holomorphic C-valued functions where K is a nonvoid compact subset of a complex metrizable locally convex space E, and ?(K) is endowed with the topology ?₀ obtained as an inductive limit of compact-open topologies (see §4). Not only Theorem 1 applies to ?(K), with E replaced by ?(K) in its statement, but also ?(K) leads to pure uniform holomorphy if and only if E is separable (see Theorem 2).     

A counterexample for a problem of Arhangel'skii concerning the product of Fréchet spaces

Using the continuum hypothesis, we give a counterexample for the following problem posed by Arhangel'skii: if X × Y is Fréchet for each countably compact regular Fréchet space Y, then is X an 〈α₃〉-space?     

Structure of modules over the stereotype algebra ℒ(X) of operators

It is well known that every module M over the algebra ?(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ? = E ? X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the stereotype algebra ? (X) of operators on X there exists a unique (up to isomorphism) stereotype space E such that M lies between two natural stereotype tensor products of E by X, $E \circledast X \subseteq M \subseteq E \odot X.$ . As a corollary, we show that if X is a nuclear Fréchet space with a basis, then each Fréchet module M over the stereotype operator algebra ?(X) can be uniquely represented as the projective tensor product of X by some Fréchet space E, $M = E \widehat \otimes X$ .     

On the Injective Tensor Product of Distinguished Frechet Spaces

An example of two distinguished Fréchet spaces E, F is given (even more, E is quasinormable and F is normable) such that their completed injective tensor product E?F is not distinguished. On the other hand, it is proved that for arbitrary reflexive Fréchet space E and arbitrary compact set K the space of E - valued continuous functions C(K, E) is distinguished and its strong dual is naturally isomorphic to ? where L¹(μ) = C(K)¹.     

The approximation property for spaces of holomorphic functions on infinite-dimensional spaces I

For an open subset U of a locally convex space E, let (H(U),τ₀) denote the vector space of all holomorphic functions on U, with the compact-open topology. If E is a separable Fréchet space with the bounded approximation property, or if E is a (DFC)-space with the approximation property, we show that (H(U),τ₀) has the approximation property for every open subset U of E. These theorems extend classical results of Aron and Schottenloher. As applications of these approximation theorems we characterize the spectra of certain topological algebras of holomorphic mappings with values in a Banach algebra.     

The ∂̄ equation in DFN spaces

We obtain: “Let E be a strong dual of a complex nuclear Fréchet space (a DFN space for short) and let F be a closed C^∞ form of type (0, 1) on E. Then there exists a C^∞ function f on E as the solution of $\text{?}\text{?}\text{f=F.”}$ Since every dual nuclear complete locally convex space may be considered (from the viewpoint of its bounded sets) as an inductive limit of DFN spaces this result is immediately applicable to problems of infinite dimensional holomorphy in a setting that goes far beyond that of DFN spaces. Furthermore this result and a lemma used in its proof improve previous of C. J. Henrich and P. Raboin on the ?? equation in Hilbert or DFN spaces.     

Holomorphic liftings and Bergman kernel estimates for 𝒟ℱ𝒩‐domains

Let E be a 𝒟ℱ𝒩‐space and let U ⊂ E be open. By applying the nuclearity of the Fréchet space ℋ︁(U) of holomorphic functions on U we show that there are finite measures μ on U leading to Bergman spaces of μ ‐square integrable holomorphic functions. We give an explicit construction for μ by using infinite dimensional Gaussian measures. Moreover, we prove boundary estimates for the corresponding Bergman kernels K_μ on the diagonal and we give an application of our results to liftings of μ ‐square integrable Banach space valued holomorphic functions over U. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Characterizations of the Properties LB^-∞ and LB∞

The aim of this paper is to give some properties of the linear topological invariant Using these results we show that a nuclear Fréchet space F has the property LB _∞ if and only if every separately holomorphic function on an open subset U × V of E × F* has a local Dirichlet representation, where E is a nuclear Fréchet space with the property having a basis.     

Weak and strong convergence of amarts in Fréchet spaces

Several new characterizations of nuclearity in Fréchet spaces are proved. The most important one states tat a Fréchet space is nuclear if and only if every mean bounded amart is strongly a.s. convergent. This extends the result in [A. Bellow, Proc. Nat. Acad. Sci. USA73, No. 6 (1976), 1798–1799] in a more positive way, and gives a different proof of it. The results of Brunel and Sucheston [C. R. Acad. Sci. Paris Ser. A (1976), 1011–1014], are extended to yield the same characterization of reflexivity of a Fréchet space in terms of weak convergence a.s. of weak amarts.     

Slice theorem and orbit type stratification in infinite dimensions

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.     

Metrizability of spaces of holomorphic functions

In this paper we prove that if U is an open subset of a metrizable locally convex space E of infinite dimension, the space H(U) of all holomorphic functions on U, endowed with the Nachbin-Coeuré topology τδ, is not metrizable. Our result can be applied to get that, for all usual topologies, H(U) is metrizable if and only if E has finite dimension.