PomeProposition 22.1.48.EDFB2682E78E Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$, $C\in \mathbb {R}$ and $\lambda ,\lambda '>0$, we have
\[ \begin{aligned} \lambda (\Gamma +\Gamma ')=& \lambda \Gamma +\lambda \Gamma ',\\ (\lambda \lambda ')\Gamma =& \lambda (\lambda '\Gamma ),\\ \lambda (\Gamma +C)=& \lambda \Gamma +\lambda C. \end{aligned} \]
Suppose that $(\Gamma ^ i)_{i\in I}$ is a non-empty family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.35). Then
\[ \lambda \left( \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i \right)=\operatorname *{sup}_{i\in I}{}\! {}^{*}(\lambda \Gamma ^ i). \]
If $(\Gamma ^ i)_{i\in I}$ is a decreasing net in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.42), then
\[ \lambda \left( \inf _{i\in I}\Gamma ^ i \right)=\inf _{i\in I}(\lambda \Gamma ^ i). \]
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