FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1995, VOLUME 1, NUMBER 1, PAGES 221-228

Locally convex modules

Z.S.Lipkina

Let K be a non-archimedean valued field, $A \subseteq K$ be its integer ring.

This paper is devoted to the study of the locally convex topological unital A-modules. These modules are very close to the vector spaces over non-archimedean valued fields. In particular, the topology of these modules can be determined by some system $\Gamma$ of semipseudonorms.

Monna demonstrated that p-adic analogue of Hahn--Banach theorem can be proved for the locally convex vector spaces over non-archimedean valued fields. One can give the definitions of q-injectivity, where q is the seminorm which is determined on this module, and of the strong topological injectivity. It means that any q-bounded homomorphism can be extended with the same seminorm, where q is a some fixed seminorm in the first case, and an arbitrary seminorm $q \in \Gamma$ in the second one.

The necessary and sufficient conditions of q-injectivity and strong topological injectivity for torsion free modules are given.

At last, the necessary and sufficient conditions for topological injectivity of a locally convex A-module in the case when A is the integer ring of the main local compact non-archimedean valued field are the following ones: a topological module is complete and Baire condition holds for any continuous homomorphism (here topological injectivity means that any continuous homomorphism of a submodule can be extended to a continuous homomorphism of the whole module).

All articles are published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/95/951/95112.htm
Last modified: October 7, 1997.