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Automatic continuity of biseparating maps. (English) Zbl 1056.46032
Let X, Y be realcompact spaces, E and F normed spaces, C(X,E) the set of all continuous E-valued functions on X and C b (X,E) the space of all bounded continuous functions from X into E. A linear map T:C(X,E)C(Y,F) is called separating if f(x)g(x)=0 for every xE implies that (Tf)(y)(Tg)(y)=0 for every yY, and biseparating if T -1 exists and is separating as well. One result of the paper is that every linear biseparating map T:C b (X,E)C b (Y,F) is continuous provided that Y has no isolated points. Another result tells us that if additionally E and F are infinite-dimensional, then a linear biseparating map T:C(X,E)C(Y,F) is continuous if the interior of the set of P-points of Y is empty. This is the best possible result.
MSC:
46E40Spaces of vector- and operator-valued functions
47B33Composition operators
46H40Automatic continuity
47B38Operators on function spaces (general)
46E25Rings and algebras of continuous, differentiable or analytic functions