Let $V$ be a vector in $\mathbb{R}^n$ and $\mathcal{T}$ an operator acting on the vector. Let $V^*$ be the fixed point of $\mathcal{T}$ such that \begin{equation} \mathcal{T}V^* = V^* \end{equation} Assume that for $\mathcal{T}$ and any $V$, $\mathcal{T}V - V^*$ is a sub-convex combination: \begin{equation} \mathcal{T}V_i - V^*_i = \sum_{k=1}^n w_k (V_k - V_k^*) \quad \forall i = 1, \:...\:n \end{equation} where $\sum_{k=1}^n w_k= \gamma \le 1$ and $w_k \ge 0$
Can I then conclude the following? \begin{equation} \vert \mathcal{T}V_i - V^*_i \vert \le \gamma \vert\vert V - V^*\vert\vert \end{equation} where $\vert\vert . \vert\vert$ is the supremum norm, i.e. $\vert\vert v \vert\vert = \sup_k \vert v_k\vert$