The algebraic equalities and their topological consequences in weighted spaces

We study algebraic equalities and their topological consequences in weighted Banach, Fréchet, or (LB) spaces of holomorphic-like functions on a locally compact and σ-compact Hausdorff space X. Our main results are the following: (1) The algebraic equality VA(X)=V0A(X) for (LB)-spaces with O- and o-growth conditions given by a weight sequence V=(vn)n always implies that these spaces are (DFS). The converse statement is valid under the additional condition (CD) which is a weakened version of the typical biduality condition for the steps Avn(X) and A(vn)0(X) generating VA(X) and V0A(X), respectively; (2) Under the same condition (CD), the algebraic equality AV¯(X)=AV¯0(X) between the projective hulls of VA(X) and V0A(X) is equivalent to AV¯(X) semi-Montel. Thus, we completely remove or significantly weaken some stringent conditions used before in many papers studying the similar problems (see, e.g., Bierstedt and Bonet, 2006 [5] and references therein).