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An example of a continuum of pairwise non-isomorphic spaces of \(C^{\infty}\)-functions. (English) Zbl 0599.46028

There is given a family \((K_{\tau})\) of compact sets \(K_{\tau}\) in the Euclidean plane with \(\tau\) ranging in a real interval such that the Whitney spaces \({\mathcal E}(K_{\tau})\) are pairwise non-isomorphic. A successful distinction of the topological structures which is sufficient for this result is managed by a certain topological property involving an increasing monotone function on \(R_+\). After Zaharyuta had presented a continuum of pairwise non-isomorphic spaces of analytic functions the open question for an analogous example in the frame of \(C^{\infty}\)- functions is clarified by a positive answer.

MSC:

46E10 Topological linear spaces of continuous, differentiable or analytic functions
46A04 Locally convex Fréchet spaces and (DF)-spaces
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