Espacios de Fréchet de generación débilmente compacta. (Weakly compactly generated Fréchet spaces) (Q1824813)

Let E be an infinite dimensional Fréchet space which is weakly compactly generated (in the sense that there exists a total absolutely convex and weakly compact subset W of E). We fix such a W and an increasing fundamental sequence \((\| \cdot \|_ n)_ n\) of continuous seminorms for E. Let \(\omega\) denote the first infinite ordinal and \(\mu\) the first ordinal whose cardinal number equals the density character d(E) of E. The author proves that there exists a resolution \(\{P_{\alpha}\); \(\omega\leq \alpha \leq \mu \}\) of the identity in E such that \(\| P_{\alpha}\|_ m=1\) and \(P_{\alpha}(W)\subset W\) for \(\omega\leq \alpha \leq \mu\), \(m=1,2,... \). (Here resolution of the identity means that \((P_{\alpha})_{\alpha}\) is an equicontinuous family of projections on E with \(P_{\mu}=id_ E\), \(P_{\alpha}\circ P_{\beta}=P_{\beta}=P_{\beta}\circ P_{\alpha}\) for \(\omega\leq \beta \leq \alpha \leq \mu\), \(d(P_{\alpha}(E))\leq | \alpha |\) for \(\omega\leq \alpha \leq \mu\) and such that for any limit ordinal \(\alpha >\omega\), \(P_{\alpha}(E)\) equals the closure of \(\cup \{P_{\eta}(E)\); \(\omega \leq \eta <\alpha \}.)\)- For Banach spaces E this result is due to \textit{D. Amir} and \textit{J. Lindenstrauss} [Ann. Math., II. Ser. 8, 35-46 (1968; Zbl 0164.149)], but the present proof is simpler and easily extends to Fréchet spaces E.