The algebraic equalities and their topological consequences in weighted spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:422

Article

Abanin, Alexander V.^1,2  Pham Trong Tien³ 

[1] Southern Inst Math, Vladikavkaz 362027, Russia

[2] Southern Fed Univ, Rostov Na Donu 344090, Russia

[3] Hanoi Univ Sci, VNU, Thanh Xuan, Ha Noi, Vietnam

关键词: Weighted function spaces;    Weighted inductive limits;    Growth conditions;   

DOI  :  10.1016/j.jmaa.2014.08.053

来源: Elsevier

【 摘 要 】

We study algebraic equalities and their topological consequences in weighted Banach, Frechet, or (LB) spaces of holomorphic-like functions on a locally compact and sigma-compact Hausdorff space X. Our main results are the following: (1) The algebraic equality VA(X) = V(0)A(X) for (LB)-spaces with O- and o-growth conditions given by a weight sequence V = (v(n))(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 A(vn)(X) and A((vn)0)(X) generating VA(X) and V(0)A(X), respectively; (2) Under the same condition (CD), the algebraic equality A (V) over bar (X) = A (V) over bar (0)(X) between the projective hulls of VA(X) and V(0)A(X) is equivalent to A (V) over bar (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). (C) 2014 Published by Elsevier Inc.

【 授权许可】

Free