We explore the assignment of norms to Λ-modules over a finite-dimensional algebra Λ, resulting in the establishment of normed Λ-modules. Our primary contribution lies in constructing two new categories \(\mathscr{N}or^{p}\) and \(\mathscr{A}^{p}\) , where each object in \(\mathscr{N}or^{p}\) is a normed Λ-module N limited by a special element υN ∈ N and a special Λ-homomorphism \(\delta_{N}:N^{\oplus2^{\text{dim}\;\Lambda}}\rightarrow{N}\) , the morphism in \(\mathscr{N}or^{p}\) is a Λ-homomorphism θ: N → M such that θ(υN) = υM and \(\theta\delta_{N}=\delta_{M}{\theta}^{\oplus2^{\text{dim}\;\Lambda}}\) , and \(\mathscr{A}^{p}\) is a full subcategory of \(\mathscr{N}or^{p}\) generated by all Banach modules. By examining the objects and morphisms in these categories, we establish a framework for understanding the categorification of integration, series expansions, and derivatives. Furthermore, we obtain the Stone-Weierstrass approximation theorem in the sense of \(\mathscr{A}^{p}\) .
