Let $K=\mathbb{Q}_p$ and $G$ a split reductive group over $K$ with split maximal torus $T$. Furthermore, $X$ is a smooth projective variety over $K$ with a free $G$-action. Let $U \subset X$ be a $T$-invariant affine subspace. Then $\mathcal{O}_X(U)$ is $K$-vector space equipped with a $T$-action. Hence, we have a decomposition into weight spaces $$ \mathcal{O}_X(U)= \bigoplus_{\lambda \in X^*(T)} \mathcal{O}_X(U)_\lambda. $$ Additionally, we equip $\mathcal{O}_X(U)$ with a norm $\lvert \, \, \rvert$ such that
$$ \lvert f \rvert =\sup_{x \in U^{\mathrm{cl}}} \{\lvert f(x)\rvert_K \} $$ for $f \in \mathcal{O}_X(U).$

Let $f \in \mathcal{O}_X(U)$ and $f = \sum f_\lambda$ with $f_\lambda \in \mathcal{O}_X(U)_\lambda$. By the properties of the absolute value on $K$, we have $$ \lvert f \rvert \leq \sup_{\lambda \in X^*(T)} \vert f_\lambda \rvert. $$

But is it true that $$ \lvert f \rvert = \sup_{\lambda \in X^*(T)} \vert f_\lambda \rvert $$ holds?