Let $(E,\mathscr E)$ be a measure space and $P:E \times\mathscr E\to [0,1]$ be a stochastic kernel - i.e. $$ P(x,A)\in [0,1] $$ for any $x\in E$ and $A\in \mathscr E$. On a set $b\mathscr E$ of bounded measurable functions with a norm $$\|f\| = \sup\limits_{x\in E}|f(x)|$$ define the action of the kernel as a linear operator $$ Pf(x) = \int\limits_E f(y)P(x,dy). $$ Let $\tilde P$ be another probability kernel and consider two norms $$ \|\tilde P - P\|' = \sup\limits_{A\in \mathscr E}\sup\limits_{x\in E}|\tilde P(x,A) - P(x,A)| $$ $$ \|\tilde P - P\|'' = \sup\limits_{f\in b\mathscr E\setminus\{0\}}\frac{\|(\tilde P - P)f\|}{\|f\|}. $$
It is easy to show that $\|\tilde P - P\|'\leq \|\tilde P - P\|''$ since an indicator function $1_A\in b\mathscr E$ for all measurable sets $A$. I wonder if the reverse inequality is true as well.
My idea was to consider a simple function $f(x) = \sum\limits_{i=1}^n f_i1_{E_i}(x)$ where $E_i$ is a partition of $E$. But if I am not missing anything, even for simple function the reverse inequality is not true.