Weak property \((Y_0)\) and regularity of inductive limits
Let \((E_n, t_n)_{n\in\mathbb{N}}\) be an inductive sequence of locally convex spaces, let \((E,t)\) be its inductive limit and assume that \((E,t)\) is Hausdorff. Then \((E,t)\) is said (a) to be regular, if each bounded set in \((E,t)\) is already contained and bounded in some space \((E_m, t_m)\), (b) to satisfy the weak property \((Y_0)\) if for each series \(\sum^\infty_{k=1} x_k\) in \((E,t)\) which is weakly unconditionally Cauchy (w.u.c.) there is \(m\in\mathbb{N}\) such that \(\sum^\infty_{k=1} x_k\) is w.u.c. in \((E_m, t_m)\). The main result of the paper is that \((E, t)\) is regular if and only if \((E, t)\) has the weak property \((Y_0)\). If all the spaces \((E_n, t_n)_{n\in\mathbb{N}}\) are sequentially complete then these properties are also equivalent to the property that for each w.u.c. series \(\sum^\infty_{k=1} x_k\) in \((E,t)\) there is \(m\in\mathbb{N}\) such that for each null-sequence \((\xi_k)_{k\in\mathbb{N}}\) of scalars, the sequence \(\sum^\infty_{k=1} \xi_kx_k\) converges in \((E_m,t_m)\). Further equivalences are derived under additional hypotheses like: No space \((E_n, t_n)\) contains a copy of \(c_0\) or all \((E_n,t_n)\) are strictly webbed spaces.
scientific article; zbMATH DE number 1496299
scientific article