The regularity of spaces of germs of Fréchet-valued bounded holomorphic functions (Q1306553)

Let \(E, F\) be two Fréchet spaces and let \(K\) be a compact set in \(E\). Then the space of germs of \(F\)-valued bounded holomorphic functions on \(K\) is denoted by \(H_{\infty}(K, F)\). Here the author shows that \(F\) has the property \((DN)\) if and only if \(H_{\infty}(K, F)\) is regular for every compact set in any Fréchet space \(E\) with the property \((\Omega)\). Moreover the Fréchet space \(E\) has the property \((\Omega)\) if and only if \(H_{\infty}(K, F)\) is regular for every compact set in any Fréchet space \(F\) with the property \((DN)\). As an application it is shown that for nuclear Fréchet spaces \(E\) and \(F\) such that \(E\in (\Omega)\) and \(F\in\overline{(DN)}\), the space \(H(K, F)\) of germs of \(F\)-valued holomorphic functions on \(K\) is regular and complete.