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Separating maps and linear isometries between some spaces of continuous functions. (English) Zbl 0918.46026
For a given locally compact Hausdorff space X, a Banach space E and a function σ:X(0,) satisfying certain conditions, the author defines the Banach space C 0 σ (X,E) of continuous functions from X into E. An additive map T:C 0 σ (X,E)C 0 τ (Y,F) between two such Banach spaces is said to be separating if whenever f,gC 0 σ (X,E) satisfy f(x)g(x)=0 for every xX, then (Tf)(y)(Tg)(y)=0 for every yY. T is said to be biseparating if it is bijective and both T and T -1 are separating. The author proves that the existence of a biseparating map T:C σ (X,E)C 0 τ (Y,F) implies that the spaces X and Y are homeomorphic.
MSC:
46E15Banach spaces of continuous, differentiable or analytic functions
46B04Isometric theory of Banach spaces