In non-locally bounded -spaces the norm is not almost transitive
The purpose of this note is to give an example of a distinguished Fréchet space and a non-distinguished Fréchet space which have the same inductive dual. Accordingly, distinguishedness is a property which is not reflected in the inductive dual. In contrast to this example, it was known that the properties of being quasinormable or having the density condition can be characterized in terms of the inductive dual of a Fréchet space.
Let be an Orlicz space defined by a convex Orlicz function and let be the space of finite elements in (= the ideal of all elements of order continuous norm). We show that the usual norm topology on restricted to can be obtained as an inductive limit topology with respect to some family of other Orlicz spaces. As an application we obtain a characterization of continuity of linear operators defined on .
Let L be a normal Banach sequence space such that every element in L is the limit of its sections and let E = ind En be a separated inductive limit of the locally convex spaces. Then ind L(En) is a topological subspace of L(E).
he paper is devoted to investigation of Gegenbauer white noise functionals. A particular attention is paid to the construction of the infinite dimensional Gegenbauer white noise measure , via the Bochner-Minlos theorem, on a suitable nuclear triple. Then we give the chaos decomposition of the L²-space with respect to the measure by using the so-called β-type Wick product.