A NOTE ON THE CONTINUITY OF MULTILINEAR MAPPINGS IN TOPOLOGICAL MODULES

In the present note, we obtain a criterion for the equicontinuity of families of multilinear mappings between topological modules. We also give an example which shows that the hypothesis imposed on the neighborhoods of zero is essential for the validity of our theorem.


INTRODUCTION.
Our goal is to establish the below generalization of a classical result of Functional Analysis ( [1], chap.I, 1, proposition 6).
THEOREM.Let A be a commutative topological ring, El, ., En, F topological A-modules and A" a family of A-multilinear mappings from E1 E, into F.If the product of any neighborhood of 0 in A by any neighborhood of O in Ei is a neighborhood of 0 in Ei(1 <_ _< n), then the equicontinuity of A" at (O, -, 0) implies the equicontinuity of,.
There are neighborhoods WH of 0 in F such that }2H WH C W. For each 1 _< _< n there is a neighborhood Ui of O in Ei so that the relations z, _ Ui imply WH for every v ,.
V(Zl, ., z,) e H e 7 Now, let Z be a neighborhood of 0 in A such that Za, C U, for every 1 < < n, and let V/ be a neighborhood of 0 in E, such that v, c unzu, nzF, n n z"-'u (1_< i_< n).

HG HG
This proves the theorem.
COROLLARY.Let A be a commutative topological ring with identity, and let n N*.If 0 E A (where A denotes the multiplicative group of all invertible elements of A), then the following property is verified: For all unitary topological A-modules El, ., E,, F and all family 2' of A-multilinear mappings from E x x E, into F, the equicontinuity of 2" at (0, , 0) implies the equicontinuity of 2".REMARK 1.By the theorem, the conclusion of the corollary remains valid if we assume that A is a commutative topological ring with the following property: For every topological A-module E, the product of any neighborhood of 0 in A by any neighborhood of 0 in E is a neighborhood of 0 in E.
Nevertheless, if A has identity, this is not a generalization.Indeed, suppose that there is a neighborhood V of 0 in A which has no invenible element.Let E be the A-module A endowed with its trivial topology (that is, the topology whose only open sets are and E).Then E is a topological A- module and the product of V by E is not a neighborhood of 0 in g.
Let us now give an example which shows that the conclusion ofthe corollary is not always true (even for non-discrete topological tings) if n _> 2. EXAMPLE.Assume n > 2. Let (A,),ez be a family of commutative tings with identity such that there is an/o I for which A is not the zero ring.Consider each A, endowed with the discrete topology, and let A be the product topological ring II, et Ai.We shall show that the conclusion of the corollary is false for the topological ring A. In order to do so, let M be the product group (Ai0) r endowed with the following law: ((a,)iei, (Xk)keN) e A M (a/0x/)keN e M.
It is easily verified that M is a unitary A-module.Let E (resp.F) be the A-module M endowed with the product topology (resp.the discrete topology).A simple argument shows that E and F are topological A-modules.Now, define Clearly, f is A-multilinear.If V { (zk) e E; z0 0}, then -o for any ((x(')), -, (x"')) in the open neighborhood V x E "-of (0,. ., 0)in E".In particular, is continuous at (0, , 0).Nevertheless, f is discontinuous.Indeed, put (:r>) (e, O, 0 0,. )and <:r('>) (:r(")) (e e, e k where e denotes the identity of A, o.Let V(1 _< j _< n) be a neighborhood of (x3)) in E. We know that V. contains a neighborhood of (x(k ") of the form _(n) for 0 < j < r} v'= (z)E;z= Now, if (ck) (X(on), ", x n>, O, O, 0,. ), then ((x(kl))) (:(kn-1)))(k)) ( V1 X X Vn_ X V C VI x X V'm and This establishes the discontinuity of f.REMARK 2. Let E and F be as in the above example, and let be the identity mapping from E into F.It is easily seen that is open, its kernel is closed and its graph is closed; however, is discontinuous.This shows, in particular, that the Open Mapping and Closed Graph Theorems are not always true in the context of complete metrizable topological modules, although they are known to be valid for a quite wide class ofcomplete metrizable topological modules ([2], corollary 12.18).