My question here is in connection with one of my previous question 
"[A definition of a (amalgamated) direct sum](https://mathoverflow.net/questions/269537/a-definition-of-a-amalgamated-direct-sum#)"
Following the notations there, my question is:


Why the locally analytic vectors of $B(V)$ is *not* isomorphic to $A(\alpha)/L^{loc}(\alpha)$
where $L^{loc}(\alpha)$ is the locally analytic vectors of $L(\alpha)$?


In general, is it true that if $W\_1/W\_2$ is a $\mathbb{Q}\_p$-Banach representation of $GL\_2(\mathbb{Q}\_p)$, then $$(W\_1/W\_2)^{loc}\cong W\_1^{loc}/W\_2^{loc},$$,


where $(-)^{loc}$ denotes the locally analytic vectors of $(-)$? 



Locally analytic vectors of a quotient space
It holds if you work in the category of admissible representations (Schneider-Teitelbaum). Otherwise it can fail. 


For a natural counterexample, see for instance my paper with Colmez "Théorie de Sen et vecteurs localement analytiques", section 3.3


