Inductive Limits Of Spaces Research Articles

Weighted inductive limits of spaces of entire functions

Methods developed by Lusky in connection with the isomorphic classification of weighted Banach spaces of entire functions are used to show that projective description holds for weighted inductive limits of spaces of entire functions for radial weights of a certain type. It is shown by an example that our results are not covered by the general theorem of Bierstedt, Meise and Summers from 1982.

The dual of the space of holomorphic functions on localy closed convex sets

Let $H(Q)$ be the space of all the functions which are holomorphic on an open neighbourhood of a convex locally closed subset~$Q$ of $\mathbb{C}^N$, endowed with its natural projective topology. We characterize when the topology of the weighted inductive limit of Frechet spaces which is obtained as the Laplace transform of the dual $H(Q)'$ of $H(Q)$ can be described by weighted sup-seminorms. The behaviour of the corresponding inductive limit of spaces of continuous functions is also investigated.

On the topological description of weighted inductive limits of spaces of holomorphic and harmonic functions

A necessary condition is given for the case, that the topology in a weighted inductive limit of spaces of holomorphic functions can be described by the canonical weighted sup-norms. This leads to an easy scheme to construct counterexamples to the problem of projective description. Similar conditions are shown, under suitable assumptions, to be necessary and sufficient in the case of harmonic functions.

Invariants of three-dimensional manifolds

O. Introduction There are several approaches to the construction of invariants of a threedimensional manifold using quantum groups, and, more generally, monoidal categories. In this paper we consider the combinatorial approach which was suggested by Turaev and Viro (see [TV] and later generalizations in [T2], [KS], [Po], [Ro] among others) and construct an invariant starting from a monoidal category C without any braiding conditions. The starting point of a combinatorial approach is a triangulation of a given three-dimensionai M, and the invariant is constructed using the combinatorics of this triangulation. More precisely, fix a monoidal category C (a category with the multiplication fimctor X ® Y on objects) over a field k, which is semisimple, has a finite number of isomorphism classes of simple objects, and possesses a duality X ~-~ X* such that X** is isomorphic to X. To define Ic(M), we must fix a balancing, i.e. functorial isomorphisms X ~ X**, X E Ob C, satisfying certain natural conditions. On the other hand, we do not assume that C satisfies any kind of braiding conditions, i.e. commutativity conditions of the form X®Y ~ Y®X. A similar construction, which also does not use braiding, was recently suggested by Kuperberg [Ku], who used the la~lguage of Hopf algebras. For any three-dimensional manifold M with boundary S and a triangulation D of M, we construct a finite-dimensional linear space W(S, D ~) depending on the restriction D ~ of D to S (a triangulation of S) and a vector It(M, S, D) C W(S, D'). The main result of the paper (Theorem 1) is that Ic(M,S,D) depends only on D ~ and not on D itself. Taking the inductive limit of spaces W(S, D ~) over all triangulations D ~ of S, we construct a space K(S) and a vector Ic(M, S) C K(S), which is our invariant of a three-dimensionM manifold M with boundary S. (See §6 of the paper, where this is explained in slightly different terms.) In particular, when M is a closed manifold, S = O, we have K(O) = k, and our construction gives an invariant Ic(M) e k.