Structure of spaces of germs of holomorphic functions (Q1385190)

Let \(E\) be a Frechet space or a Frechet-Hilbert space. If the topology of \(E\) is defined by the increasing fundamental system of seminorms \(\{\| \cdot \|_k\}_{k =1}^\infty\) then for each subset \(B\) of \(E\) \[ \| u\|^*_B =\sup \biggl\{\bigl| u(x) \bigr |: x\in B\biggr\} \] and \(\| \cdot\|^*_ q=\| \cdot \|^*_B\) if \(B= \{x\in E: \| x \|_q \leq 1\}\). Following Vogt the authors introduce the following two properties of the space \(E\) \[ (DN): \exists p\forall q \exists k,\;C>0:\|\cdot\|^2_q \leq C\|\cdot\|_k\|\cdot\|_p; \] \[ (\Omega): \forall p\exists q\forall k \exists d,\;C>0: \| \cdot\|^{*(1+d)}_q \leq C\|\cdot\|^*_k\|\cdot\|_p^{*d}. \] They prove several theorems given necessary and sufficient conditions of possession of these properties for the space \(E\).