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.


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".

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