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.     

An example of a nuclear Fréchet space without the bounded approximation property

Mathematische Zeitschrift -     

Equilibria in abstract economies with a measure space of agents and with an infinite dimensional strategy space

Khan's result [7, 8] on equilibrium points of non-atomic games is generalized to a setting where agents' preferences need not be ordered.     

An example of a universal Banach space

We construct a separable reflexive Banach spaceX which is complementably universal for all finite dimensional Banach spaces. By this we mean: for every finite dimensional Banach spaceE there is isometric embeddingi:E→X such that there exists a projectionP: → _→ ^onto with ‖P‖=1.     

An infinite dimensional functional calculus in Banach algebras with a modulus

The presence of a modulus function on a Banach algebra gives rise to tensor product norms analogous to the greatest cross‐norm and allows us to extend the Waelbroeck functional calculus. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An infinite dimensional Morse theory with applications

In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the one-dimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations.

     

An infinite dimensional version of the Paley-Wiener-Schwartz isomorphism

In the infinite dimensional case it is known from a Dineen-Nachbin counterexample that the natural generalization of the Paley-Wiener-Schwartz theorem is no longer valid: The image through the Fourier transform F of the space ?′(E) is made of entire functions on E′ that, besides the usual inequality, satisfy a more technical condition. We define and study here a dense subspace, denoted by E(E), of ?(E) with a proper topology such that F (E′(E)) is made of the entire functions on E′ that satisfy only the classical inequality. For this reason this space E(E) is suited for the study of convolution and partial differential equations in spaces of C^∞ functions on locally convex spaces.     

A comparison result for elliptic equations in the infinite dimensional Gauss space

We obtain a comparison result for a class of Dirichlet problems for the operator in an infinite dimensional separable Hilbert space X with the Gauss measure γ and a suitable differentiable structure.     

An infinite dimensional Schur-Horn Theorem and majorization theory

The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences ξ and η that converge to 0, there exists a positive compact operator A with eigenvalue list η and diagonal sequence ξ if and only if for every n if and only if ξ=Qη for some orthostochastic matrix Q. When ξ and η are summable, requiring additionally equality of their infinite series obtains the same conclusion, extending a theorem by Arveson and Kadison. Our proof depends on the construction and analysis of an infinite product of T-transform matrices.     

An example of L-fuzzy join space

On a generalized deMorgan lattice (X, ≤, ∨, ∧,′) we introduce a family of join hyperoperations * _p , parametrized by a parameterp εX. As a result we obtain a family of join spaces (X, * _p ). We show that: for everya,b εX the family {a*_pb} _pεX can be considered as thep-cuts of aL-fuzzy seta*b; in this manner we synthesize aL-fuzzy hyperoperation * which takes pairs fromX toL-fuzzy subsets ofX. We then show that (X, * _p ) is aL-fuzzy hypergroup (in the sense of Corsini) and can be considered as aL-fuzzy join space. Furthermore,a*b is aL-fuzzy interval for alla,b εX.     

An example of nonexistence of a steiner point in a Banach space

For each n = 3, 4, …, we construct an example of a Banach space X and elements x ₁, …, x _n in this space such that X does not have any element with the minimal sum of distances to the elements x _k .