Let $v$ be a Henselian Krull valuation of a field $K$. In this paper, the authors give some necessary and sufficient conditions for a finite simple extension of $(K,v)$ to be defectless. Various characterizations of algebraically maximal valued fields are also given which lead to a new proof of a result proved by Yu. L. Ershov.

The study carried out in this paper about some new examples of Banach spaces, consisting of certain valued fields extensions, is a typical non-archimedean feature. We determine whether these extensions are of countable type, have $t$-orthogonal bases, or are reflexive. As an application we construct, for a class of base fields, a norm $\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm, without non-zero vectors that are $\|\cdot\|$-orthogonal and such that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$ into a valued field.