PomeF0C01F4F75EBLemma 16.9.41.F0C01F4F75EB Let $(\varphi _ i)_{i\in I}$ be a net in $\mathrm{PSH}(X,\theta )$, and $\varphi \in \mathrm{PSH}(X,\theta )$. Fix $\lambda >0$. Then the following are equivalent:
9CF34F7033CA$\varphi _ i\xrightarrow {d_{S,\theta }}\varphi $;
1FA7990ED285$\lambda \varphi _ i\xrightarrow {d_{S,\lambda \theta }}\lambda \varphi $.
Proof. We have $V_{\lambda \theta }=\lambda V_{\theta }$. The middle expression in (16.9.10) for the pair $(\lambda \varphi _ i,\lambda \varphi )$ in the class $\lambda \theta $ is therefore $\lambda ^ n$ times the corresponding expression for $(\varphi _ i,\varphi )$ in the class $\theta $. The equivalence follows from corollary 16.9.9. $\square$
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