A geometric characterization for the property \((DN)\) of \(\mathcal{E}(K)\) for arbitrary compact subsets \(K\) of \(\mathbb{R}\) (Q5953114)

A compact subset \(K\) of \(\mathbb{R}\) is said to be perfect of class \(\alpha\geq 1\) if there are \(C\geq 1\) and \(\delta> 0\) such that for each \(y\in K\) there exists a sequence \((x_j)_{j\in \mathbb{N}}\) in \(K\) with the following properties: \(|x_1- y|\geq \delta\), \(|x_j- y|\) decreases to zero, and \(|y-x_n|^\alpha\leq C|y- x_{j+1}|\) for each \(j\in\mathbb{N}\). Denote by \({\mathcal E}(K)\) the Fréchet space of all Whitney jets on \(K\). The main result of the present paper shows that \({\mathcal E}(K)\) has the linear topological invariant \((DN)\) if and only if \(K\) is perfect of some class \(\alpha\geq 1\). Note that by a result of Vogt, the property \((DN)\) characterizes the nuclear Fréchet spaces which are isomorphic to subspaces of nuclear stable power series spaces of finite type.