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**Differences of functions in locally convex spaces and applications to almost periodic and almost automorphic functions.**
*(English)*
Zbl 0536.43014

Let f be a function from a topological group G to an Abelian topological group A and let the difference \(\Delta_{\gamma}f(t)=f(t\gamma)-f(t)\) be continuous at the unity e of G for each \(\gamma\) belonging to a neighbourhood E of e.

The authors point out some conditions under which f becomes continuous at e or generally on G. Thus they prove that if either A is a Fréchet space which does not contain any subspace isomorphic to the Banach space \(c_ 0\) of convergent to zero complex sequences, or A is a locally convex space and the range \(\{y;y=f(x),x\in E\}\) is weakly compact, then if f is bounded on E, f is continuous at e. Some particular cases of this problem have already been studied by N. G. Bruijn [Nieuw. Arch. Wiskunde, II. Ser. 23, 194-218 (1951)], F. W. Caroll (Trans. Am. Math. Soc. 102, 284-292 (1962)] and H. Günzler [Math. Z. 102, 253-287 (1967; Zbl 0185.220)]. Restricting their results to the case where G is a totally bounded topological group, the authors obtain some interesting applications to the case of almost automorphic and almost periodic functions. They also obtain a result concerning the problem of the differences for left (right) uniformly continuous functions even when G is not totally bounded.

The authors point out some conditions under which f becomes continuous at e or generally on G. Thus they prove that if either A is a Fréchet space which does not contain any subspace isomorphic to the Banach space \(c_ 0\) of convergent to zero complex sequences, or A is a locally convex space and the range \(\{y;y=f(x),x\in E\}\) is weakly compact, then if f is bounded on E, f is continuous at e. Some particular cases of this problem have already been studied by N. G. Bruijn [Nieuw. Arch. Wiskunde, II. Ser. 23, 194-218 (1951)], F. W. Caroll (Trans. Am. Math. Soc. 102, 284-292 (1962)] and H. Günzler [Math. Z. 102, 253-287 (1967; Zbl 0185.220)]. Restricting their results to the case where G is a totally bounded topological group, the authors obtain some interesting applications to the case of almost automorphic and almost periodic functions. They also obtain a result concerning the problem of the differences for left (right) uniformly continuous functions even when G is not totally bounded.

Reviewer: A.Precupanu

### MSC:

43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |

11F03 | Modular and automorphic functions |