Given a vector $Q \in \mathbb{R}^{S\times A}$ where $S$ and $A$ are sets of finite cardinality, for $0<\gamma<1$ define the function $H_{w}:\mathbb{R}^{S\times A} \rightarrow \mathbb{R}^{S\times A}$ as $H_{w}Q(s,a) := w\left(r(s,a)+ \gamma \sum_{s' \in S}p(s'|s,a)\displaystyle\max_{a \in A} Q(s',a)\right)+(1-w)\displaystyle\max_{a \in A}Q(s,a).$ Here $|r(s,a)| \leq R ~ \forall ~ (s,a) \in S \times A$ and $p(\_|s,a)$ is a probability mass function on $S$ for any given $(s,a) \in S \times A$ and $0<w<w^*$ with $w^*=\displaystyle\max_{(i,a)}\left\{\frac{1}{1-p(i|i,a)}\right\}.$

Note:

- $H_w$ is a contraction with respect to the max-norm/infinity-norm on $\mathbb{R}^{S\times A}$ with contraction factor $(1-w+\gamma w)$
- Note the possibility of $w>1$.

Is the function $H_wQ$ jointly Lipschitz in $w$ and $Q$?