I want to find the Lipschitz constant of the function $g(Bx)$, where $g:\mathbb{R}^{m} \rightarrow \mathbb{R}$ is a K-Lipschitz function, $B \in \mathbb{R}^{m \times n}$ is a matrix, and $x \in \mathbb{R}^{n}$ is fixed.
First consider a function $f(B) = Bx$. We can determine its Lipschitz constant as follows \begin{equation*} L = \max_{B_1 \neq B_2} \frac{\|B_1 x - B_2 x\|}{\| B_1 - B_2\|} = \max_{B \neq 0} \frac{\|B x\|}{\| B\|} = \|x\| \end{equation*}
Using this result, I can define the Lipschitz norm of $g(Bx)$ as $K \|x\|$. I was wondering when $B$ is a vector instead of a matrix, $B = y^\top$, I will still get $\|x\|$ using the above reasoning. However, Cauchy-Schwartz inequality suggests that the $L = \|x\|_{*}$, where $\|\|_{*}$ is the dual norm. So the question boils down to, if the following submulticative property is true, $\|B x\| \stackrel{?}{\leq} \|B\| \|x\|_{*}$.