Completeness of \(L^1\)-spaces for measures with values in complex vector spaces (Q1269592)

A direct proof of a theorem is given stating that if \(X\) is a complex Fréchet space and \(\nu:\Sigma\to X\) is a vector measure, then the space \(L^1(\nu,X)\) is \(\tau(\nu)\)-complete and thus it is a Fréchet space. Moreover, it is proved that if \(X\) is a complex sequentially complete, locally convex Hausdorff space, \(\nu: \Sigma\to X\) -- a vector measure, then \(L^1(\nu, X)\) is complete if and only if \(\nu\) is a closed measure.