Biström, Peter; Jaramillo, Jesús A. \(C^ \infty\)-bounding sets and compactness. (English) Zbl 0824.46033 Math. Scand. 75, No. 1, 82-86 (1994). 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 \(f\in C^ \infty(E)\). Reviewer: J.A.Jaramillo (Madrid) Cited in 1 Document MSC: 46E25 Rings and algebras of continuous, differentiable or analytic functions Keywords:\(C^ \infty\)-bounding sets; compactness PDFBibTeX XMLCite \textit{P. Biström} and \textit{J. A. Jaramillo}, Math. Scand. 75, No. 1, 82--86 (1994; Zbl 0824.46033) Full Text: DOI EuDML