On strong basis (Q1281119)

A Schauder basis \(\{x_i, f_i\}\) in a locally convex space \(E\) is said to be semi-absolute if for each \(x\in E\) and each continuous seminorm \(p\) on \(E\), \((f_i(x) p(x_i))_i\in \ell^1\) and a strong basis if for each absolutely convex closed and bounded set \(B\) in \(E\) and each continuous seminorm \(p\) on \(E\), \((p_B(f_i) p(x_i))_i\in \ell^1\). The author shows that for a locally convex space \(E\) with weakly sequentially complete dual, a Schauder basis \(\{x_i, f_i\}\) is a strong basis if and only if it is semi-absolute and \(\{f_i, Jx_i\}\) is a semi-absolute basis for the strong dual of \(E\). A Schauder basis \(\{x_i, f_i\}\) in a Fréchet space \(E\) is a strong basis if and only if \(\{f_i, Jx_i\}\) is a strong basis for \(E_b'\). By a result of \textit{U. Mertins} [Stud. Math. 48, 223-231 (1973; Zbl 0264.46003)], this holds if and only if \(E\) is nuclear. Any Schauder basis in a nuclear (DF)-space is a strong basis, and conversely, if \(E\) is a barrelled space with a fundamental sequence of bounded sets and a strong basis, \(E\) must be nuclear.