This chapter is basic for the whole monograph. Let k ⊂ K be a field extension. We study the category of pairs consisting of a finite-dimensional vector space V over k and a filtration on the K-vector space V ⊗kK. This is a quasi-abelian k-linear tensor category, whose set of objects is naturally endowed with two ℝ-valued functions, called degree and rank, which are additive on short exact sequences. The quotient of degree by rank is called slope. It is additive on tensor products and convex on short exact sequences. There is here a strong analogy with the k-linear quasi-abelian category of vector bundles over a projective smooth curve over k, endowed with the usual degree and rank functions. Keeping this analogy in mind, we introduce the notion of semi-stable objects, and show how any object carries a canonical filtration with semi-stable and slope-decreasing subquotients, called its Harder–Narasimhan filtration. Making this analogy explicit is not merely a pleasant exercise in linear algebra. It is motivated by Hodge theory, complex or p-adic, as explained in the general introduction of this monograph. Technically, the most difficult result is the tensor product theorem of Faltings and Totaro, which essentially states that the canonical filtration of a tensor product is the tensor product of the respective canonical filtrations. The importance of this theorem for the subject matter of this monograph will become apparent in Part 2.