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.     

Productively Fréchet Spaces

We solve the long standing problem of characterizing the class of strongly Fréchet spaces whose product with every strongly Fréchet space is also Fréchet.     

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....     

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.     

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.     

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     

Block sequences in nuclear Fréchet spaces with basis

It is shown that a block sequence in a nuclear Fréchet space with a basis has a block extension if and only if the subspace it generates is complemented. In addition, a short proof is given of the following result of Dubinsky and Robinson: a nuclear Fréchet space is isomorphic to = R^N, N = {1,2,...} if it has a basis such that any block sequence with blocks of length 2 of any permutation of this basis has a block extension. It is shown that a similar result holds without considering permutations of the basis if the length of the blocks is arbitrary.Translated from Matematicheskie Zametki, Vol. 17, No. 6, pp. 899–908, June, 1975.The author wishes to thank B. S. Mityagin for calling his attention to this problem and for his valuable suggestions.     

Asymptotic Behavior of Gaussian Samples in Fréchet Spaces

Abstract

In this article we investigate unnormalized samples of Gaussian random elements in a separable Fréchet space 𝕄. First we describe a connection between shifts of a Gaussian measure μ in a separable Fréchet space and the infinite product of standard normal distributions in ?^∞, and on the basis of this result we derive the so‐called self‐sufficient expansion for Gaussian random elements in a Fréchet space. Moreover, we find lower bounds for the Gaussian measure μ of shifted balls in 𝕄 and estimate the metric entropy of balls in the Hilbert space ? ? 𝕄 which generates μ. Finally, applying the Brunn–Minkowski inequality we prove a kind of the logarithmic law of large numbers. The last result is an extension of the analogous theorem obtained by Goodman (Characteristics of normal samples. Ann. Probab. 1988, 16, 1281–1290), for a sequence of Gaussian random elements in a separable Banach space.     

A generalized vector-valued variational principle in Fréchet spaces

In the framework of Frechet spaces, we give a generalized vector-valued Ekeland＇s variational principle, where the perturbation involves the subadditive functions of countable generating semi-norms. By modifying and developing the method of Cammaroto and Chinni, we obtain a density theorem on extremal points of the vector-valued variational principle, which extends and improves the related known results.     

Separable Reduction in the Theory of Fréchet Subdifferentials

We prove a separable reduction theorem for the Fréchet subdifferential that contains all earlier results of that sort as particular cases.     

Coorbit Spaces with Voice in a Fréchet Space

We set up a new general coorbit space theory for reproducing representations of a locally compact second countable group G that are not necessarily irreducible nor integrable. Our basic assumption is that the kernel associated with the voice transform belongs to a Fréchet space \(\mathcal T\) of functions on G, which generalizes the classical choice \(\mathcal T=L_w^1(G)\). Our basic example is \( \mathcal T=\bigcap _{p\in (1,+\infty )} L^p(G)\), or a weighted versions of it. By means of this choice it is possible to treat, for instance, Paley-Wiener spaces and coorbit spaces related to Shannon wavelets and Schrödingerlets.     

On Fréchet–Hilbert algebras

We consider Hilbert algebras with a supplementary Fréchet topology and get various extensions of the algebraic structure by using duality techniques. In particular we obtain optimal multiplier-type involutive algebras which in applications are large enough to be of significant practical use. The setting covers many situations arising from quantization rules, as those involving square-integrable families of bounded operators     

Fréchet algebras of finite type

The main objects of study in this paper are Fréchet algebras having an Arens Michael representation in which every Banach algebra is finite dimensional. We shall classify these algebras using a theorem which ensures that the image of any continuous linear map of a Fréchet space of finite type (i.e., for which the defining seminorms have a finite dimensional cokernel) into any Fréchet space is in fact closed.This work is part of the research project of the European Research Training Network Analysis and Operators, contract HPRN-CT 2000 00116, funded by the European Commission.     

Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface

Let $X$ be a compact connected Riemann surface and $G$ a connected reductive complex affine algebraic group. Given a holomorphic principal $G$ -bundle $E_G$ over $X$ , we construct a $C^\infty $ Hermitian structure on $E_G$ together with a $1$ -parameter family of $C^\infty $ automorphisms $\{F_t\}_{t\in \mathbb R }$ of the principal $G$ -bundle $E_G$ with the following property: Let $\nabla ^t$ be the connection on $E_G$ corresponding to the Hermitian structure and the new holomorphic structure on $E_G$ constructed using $F_t$ from the original holomorphic structure. As $t\rightarrow -\infty $ , the connection $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . In particular, as $t\rightarrow -\infty $ , the curvature of $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the curvature of the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . The family $\{F_t\}_{t\in \mathbb R }$ is constructed by generalizing the method of [6]. Given a holomorphic vector bundle $E$ on $X$ , in [6] a $1$ -parameter family of $C^\infty $ automorphisms of $E$ is constructed such that as $t\rightarrow -\infty $ , the curvature converges, in $C^0$ topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.     

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.     

Real analytic curves in Fréchet spaces and their duals

The following results are presented: 1) a characterization through the Liouville property of those Stein manifoldsU such that every germ of holomorphic functions on xU can be developed locally as a vector-valued Taylor series in the first variable with values inH(U); 2) ifT is a surjective convolution operator on the space of scalar-valued real analytic functions, one can find a solutionu of the equationT u=f which depends holomorphically on the parameterz wheneverf depends in the same manner. These results are obtained as an application of a thorough study of vector-valued real analytic maps by means of the modern functional analytic tools. In particular, we give a tensor product representation and a characterization of those Fréchet spaces or LB-spacesE for whichE-valued real analytic functions defined via composition with functionals and via suitably convergent Taylor series are the same.