Structural Hierarchy of Reid Class of non-Archimedean Banach Spaces

Let $k$ be a complete valuation field. We formulate a class $\mathcal{R}$ of Banach $k$-vector spaces analogous to Reid class of Abelian groups. We formulate an analogue of the hierarchy of Reid class introduced by K.\ Eda, and verify a counterpart of the classification theorem of Reid class by K.\ Eda. As an application, we verify that the Banach ${\mathbb{C}}_p$-vector spaces \begin{eqnarray*} & & \ell^{\infty}($\mathbb{N}$,$\mathbb{C}$_p), $\text{C}$_0($\mathbb{N}$,$\mathbb{C}$_p), \ell^{\infty}($\mathbb{N}$,$\text{C}$_0($\mathbb{N}$,$\mathbb{C}$_p)), $\text{C}$_0($\mathbb{N}$,\ell^{\infty}($\mathbb{N}$,$\mathbb{C}$_p)), \ & & \ell^{\infty}($\mathbb{N}$,$\text{C}$_0($\mathbb{N}$,\ell^{\infty}($\mathbb{N}$,$\mathbb{C}$_p))), $\text{C}$_0($\mathbb{N}$,\ell^{\infty}($\mathbb{N}$,$\text{C}$_0($\mathbb{N}$,$\mathbb{C}$_p))), \end{eqnarray*} and so on are all distinct, the Banach ${\mathbb{C}}_p$-vector space of bounded continuous functions $\mathbb{Q}\to {\mathbb{C}}_p$ and its dual Banach ${\mathbb{C}}_p$-vector spaces cannot be expressed by iterated application of bounded direct product and completed direct sum, and there is no left adjoint functor of the forgetful functor from $\mathcal{R}$ to the category of Banach ${\mathbb{C}}_p$-vector spaces.