×

\(C^ \infty\)-bounding sets and compactness. (English) Zbl 0824.46033

It is proved that any set on a real Banach space on which the \(C^ \infty\)-functions are bounded, is relatively compact. In particular, for any real Banach space \(E\), a sequence \((x_ n)\) converges to \(x\) in \(E\) if and only if \(f(x_ n)\) converges to \(f(x)\) for every \(f\in C^ \infty(E)\).

MSC:

46E25 Rings and algebras of continuous, differentiable or analytic functions
PDF BibTeX XML Cite
Full Text: DOI EuDML