Here, page $127$, theorem $2.12$, states the following:
Let $X$ be a Banach space. Then $\mathcal{F}(X)$ is Lipschitz-isomorphic to the space $\ker (\beta) \oplus X$. Moreover , these two spaces are linearly isomorphic if and only if the space $X$ has the lifting property.
In the proof, the author defined the following map $$L:\mathcal{F}(X) \rightarrow \ker (\beta) \oplus X$$ given by $L(\mu)=(\mu-\delta_X\beta_X(\mu), \beta_X(\mu))$. Then the author claimed that the map is a Lipschitz-isomorphism. I manage to show that the map is indeed bijection, but I have trouble showing it is Lipschitz.
In particular, I don't know what is the norm of $\ker (\beta) \oplus X$. Can anyone enlighten me?
UPDATE: Other than using the definition of Lipschitz, is there any other results that we can use to show the map $L$ is Lipschitz?
On the direct sum $\ker \beta \oplus X$, one usually uses any of the (equivalent!) norms $$ \def\norm#1{\left\|#1\right\|}\norm{(\mu, x)}_p := \bigl(\norm{\mu}^p + \norm{x}^p\bigr)^{1/p} $$