Characterizations of nuclearity in Fréchet spaces

We extend the result of A. Bellow (Proc. Nat. Acad. Sci. USA73, No. 6 (1976), 1798–1799) on the characterization of finite-dimensional Banach spaces, to a characterization of nuclearity for Fréchet spaces. Those spaces are nuclear iff every Pettis-bounded and Pettis-uniformly integrable amart is mean convergent. Several other characterizations are given.     

Double convergence and products of Fréchet spaces

The paper is devoted to convergence of double sequences and its application to products. In a convergence space we recognize three types of double convergences and points, respectively. We give examples and describe their structure and properties. We investigate the relationship between the topological and convergence closure product of two Fréchet spaces. In particular, we give a necessary and sufficient condition for the topological product of two compact Hausdorff Fréchet spaces to be a Fréchet space.     

Resolutions of the identity in Fréchet spaces

It is a classical result that every Bade -complete Boolean algebra of (selfadjoint) projections in a separable Hilbert space coincides with the projections forming the resolution of the identity of some bounded selfadjoint operator. This result is extended to the setting of separable Fréchet spaces. Namely, every Bade -complete Boolean algebra of projections in such a space coincides with the resolution of the identity of some (continuous) scalar-type spectral operator having spectrum a compact subset of.     

Intersections of Fréchet Schwartz spaces and their duals

In this article the structure of the intersections of a Fréchet Schwartz space F and a (DFS)-space E=ind _n E _n is investigated. A complete characterization of the locally convex properties of E ⋃ F is given. This space is boraological if and only if the inductive limit E + F is complete. The results are based on recent progress on the structure of (LF)-spaces. The article includes examples of (FS)-spaces F and (DFS)-spaces E such that there are sequentially continuous linear forms on E ⋃ F which are not continuous, thus answering a question of Langenbruch. Acknowledgement: The results in this article were obtained during the author’s stay at the University of Paderborn, Germany, during the academic year 1994/95. The support of the Alexander von Humboldt Stiftung is greatly appreciated. The content of the article was presented as an invited paper in a Special Session of the AMS meeting in New York in April, 1996.     

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.     

Inverse mapping theorem in Fréchet spaces

Journal of Optimization Theory and Applications - We consider the classical inverse mapping theorem of Nash and Moser from the angle of some recent development by Ekeland and the authors....     

Fréchet approach in second-order optimization

We state a certain second-order sufficient optimality condition for functions defined in infinite-dimensional spaces by means of generalized Fréchet’s approach to second-order differentiability. Moreover, we show that this condition generalizes a certain second-order condition obtained in finite-dimensional spaces.     

Geometric properties of Fréchet spaces and selection of basis sequences

For an arbitrary nuclear Fréchet spaceE possessing the well-known geometric propertiesD ₁ (DN) and Ω, certain sequences of functionals and elements with power-type estimates of the norms are selected. With the help of these objects, two isomorphic spacesK ₁ andK ₂ of power series of infinite type and continuous mapsJ ₂: K₂→E andJ ₁: E→K₁ are defined. Under some auxiliary conditions, it is proved that the_i are isomorphisms and the spaceE possesses a basis. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 102–111, July, 1999.     

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?     

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.     

Sufficient sets in weighted Fréchet spaces of entire functions

Under study are sufficient sets in Fréchet spaces of entire functions with uniform weighted estimates. We obtain general results on the a priori overflow of these sets and introduce the concept of their minimality. We also establish necessary and sufficient conditions for a sequence of points on the complex plane to be a minimal sufficient set for a weighted Fréchet space. Applications are given to the problem of representation of holomorphic functions in a convex domain with certain growth near the boundary by exponential series.     

Fixed point theory in Fréchet spaces for Volterra type operators

New fixed point theorems for maps (single and multivalued) between Fréchet spaces are presented. The proof relies on fixed point theory in Banach spaces and viewing a Fréchet space as the projective limit of a sequence of Banach spaces.     

Fréchet Means for Distributions of Persistence Diagrams

Given a distribution \(\rho \) on persistence diagrams and observations \(X_{1},\ldots ,X_{n} \mathop {\sim }\limits ^{iid} \rho \) we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams \(X_{1},\ldots ,X_{n}\) . If the underlying measure \(\rho \) is a combination of Dirac masses \(\rho = \frac{1}{m} \sum _{i=1}^{m} \delta _{Z_{i}}\) then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from \(\rho \) . We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.     

Fréchet-Valued Real Analytic Functions on Fréchet Spaces

The aim of this paper is to establish the equivalence between the real and topologically real analyticity of Fréchet-valued functions on Fréchet spaces. This is an extension of recent results of Bonet and Domanski [1, 2] to infinite dimension.Received February 19, 2002; in revised form June 25, 2002 Published online May 16, 2003