In their book "Non-Archimedean analysis", when BGR refer to an epimorphism in the context of $k$-Banach algebras do they actually require (or does it follow that) such maps are surjections? For example, after their definition of $k$-affinoid algebras, they seem to use Banach's Open mapping theorem to deduce such algebras are quotients of Tate algebras, however that would require surjectivity.



Terminology: Epimorphism in non-archimedean analysis
The authors mean a surjective homomorphism, "epimorphism" is too weak. An affinoid $k$-algebra is defined to be a Banach $k$-algebra of the form $k\langle x\_1,\dotsc,x\_n \rangle/\mathfrak{a}$ for some closed ideal $\mathfrak{a}$. This is analogous to affine $k$-algebras which have the form $k[x\_1,\dotsc,x\_n]/\mathfrak{a}$. So basically polynomials are replaced by strictly convergent power series.


