p-adic representations of $GL_2(\mathbb{Q}_p)$

Question

Let $L$ be a finite extension of $\mathbb{Q}_p$. Colmez defines here

the trainguline representations which are extensions of Robba rings of dimension $1$. Then, in this paper he contructs the representations of $GL_2(\mathbb{Q}_p)$ which corresponds to the trianguline representations.

More precisely, Let $s=\{\delta_1,\delta_2,\mathcal{L}\}$ be as defined in $0.2$ of loc.cit where $\delta_1,\delta_2$ are two characters from $\mathbb{Q}_p^{\times}$ to $L^{\times}$ and $\mathcal{L}$ is either $\infty$ or $\mathbb{P}^1(L)$. Then for each such $s$, Colmez constructs, in $0.3$, a Banach representation of $GL_2(\mathbb{Q}_p)$ which he writes by $\Pi(s)=B(s)/\widehat{M(s)}$ where $B(s)$ is some set of functions $\phi:\mathbb{Q}_p \rightarrow L$ in class $\mathcal{C}^{u(s)}$ (where $u(s)=val_p(\delta_1(p))$) with the property that the function $x \rightarrow \delta_s(x)\phi(1/x)$ can be embedded into a function of class $\mathcal{C}^{u(s)}$ at $0$ (where $\delta_s$ is the character $(x|x|)^{-1}\delta_1\delta_2^{-1}$). I know that $\Pi(s)$ is a Banch space. My question is whether $B(s)$ is a Banach space? If yes, what is the norm?

I am asking this question because in general for a $(\varphi,\Gamma)$ module $D$, when Colmez constructs (ref. this) the corresponding $GL(2)$ representation $\Pi(D)=D\boxtimes\mathbb{P}^1/D^{\natural} \boxtimes \mathbb{P}^1$, then $\Pi(D)$ is a Banach space but $D\boxtimes\mathbb{P}^1$ is not a Banach space.

Any help is welcome. Thanks in advance.

Answer 1

Apparently you didn't read the references you quoted terribly carefully, since exactly this question is addressed by the footnote on the bottom of page 4 of the paper you link:

"L'application qui a $\phi \in B(s)$ associe [...] est une isomorphisme de $B(s)$ sur [...], ce qui munit $B(s)$ d'une structure de L-banach".

Translation: "The map which, to $\phi \in B(s)$, associates [something] is an isomorphism from $B(s)$ onto [some other space], which endows $B(s)$ with the structure of an $L$-Banach space".