The structure of type \((\Omega)\) of spaces of Banach-valued holomorphic germs (Q5954626)

For a compact subset \(K\) of a Fréchet space \(E\) and a Banach space \(X\) denote by \(H(K,X)\) the space of all germs of \(X\)-valued holomorphic functions on \(K\). \(H(K,X)\) is endowed with the inductive limit topology of the family \(H^\infty(U,X)\), \(U\) any open neighborhood of \(K\). As main result of the present paper, the authors show that \(H(K,X)'\), the strong dual of \(H(K,X)\), is quasinormable (resp. has the property \((\Omega))\) if \(E\) is quasinormable (resp. \(E\) has \((\Omega))\).