Let $L$ be a line bundle on a paracompact strictly $K$-analytic space $V$. Let $x\in V$ and let ${\|\hskip 4.30554pt\|}$ be a piecewise ${\mathbb{Q}}$-linear metric on $L$.

• (a)

  The set of points in $V$ where ${\|\hskip 4.30554pt\|}$ is semipositive is open in $V$.

• (b)

  The tensor product of two piecewise ${\mathbb{Q}}$-linear metrics which are semipositive in $x$ is again semipositive in $x$.

• (c)

  Let $f:V^{\prime}\to V$ be a morphism of paracompact strictly $K$-analytic spaces. If ${\|\hskip 4.30554pt\|}$ is semipositive in $x$, then $f^{*}{\|\hskip 4.30554pt\|}$ is semipositive in any point of $f^{-1}(x)$.