Locally octahedral and locally almost square K\"othe-Bochner spaces

Abstract

It has been proved in [J.-D. Hardtke, J. Math. Phys. Anal. Geom. 16, no.2, 119--137 (2020)] that a K\"othe-Bochner space $E(X)$ is locally octahedral/locally almost square if $X$ has the respective property and the simple functions are dense in $E(X)$. Here we show that the result still holds true without the density assumption. The proof makes use of the Kuratowski-Ryll-Nardzewski Theorem on measurable selections.