Matrix Transformations and Scale Separation in Non-Archimedean λ-Almost Convergent Sequence Spaces

Let F be a complete, non-trivially valued, non-Archimedean field of characteristic zero. This paper studies the space V^λ of λ-almost convergent sequences over F and its paranormed generalization V^λ(p), establishing it as a complete non-Archimedean topological vector space for every admissible λ. The classical inclusion c⊂V^λ holds if and only if F has residue characteristic zero and m−λm is unbounded. The matrix class (c,V^λ) is characterized by means of an ultrametric gliding hump argument. The strong almost space [V^,λ] and the almost λ-statistical space s^λ coincide on l∞, yet over Qp this common space is not contained in V^λ, a separation that reflects the arithmetic of the base field.