Domains of analyticity and domains of analytic existence in real locally convex spaces
by
Murielle Mauer
University of Liège

The problem of the characterisation of domains of analyticity and domains of analytic existence has been investigated in many complex infinite dimensional spaces such as Banach spaces, Fréchet spaces and inductive limits of Silva. In the case of real spaces, Schmets and Valdivia solved the problem for normed spaces in 1993 by proving that in a real separable normed space, every non void domain is a domain of analytic existence.

Using their result, I next proved that in a countable inductive limit of real separable Banach spaces, every non void domain is still a domain of analytic existence. I investigated finally the case of a real separable Fréchet space and the results I got are the following:

- If E is a real separable Fréchet space, every non void domain is a domain of analyticity if and only if the Fréchet space has a continuous norm.

- In a real separable Fréchet space, every non void domain which is open for one semi-norm is a domain of analytic existence.

- In a real separable quojection, the converse of the last statement is true and a non void domain is a domain of analytic existence if and only if it is open for one semi-norm.

Date received: March 29, 2001