Abstract
This study develops and practically reformulates the Trotter–Kato theorem for the approximation of bi-continuous semigroups defined on Banach spaces with weaker locally convex Hausdorff topologies. This method extends the Trotter–Kato framework to semigroups with weak continuity by moving the analysis from the classical norm topology to a weaker topology. Equivalence and approximation conditions are derived without requiring explicit computation of resolvents, replacing them with T-stability and T-consistency properties. This new reformulation of the Trotter–Kato theorem provides a direct method for establishing convergence between approximating and limit semigroups, thereby simplifying both theoretical proofs and numerical analysis. The suggested framework is illustrated for diffusion equations on unbounded domains with non-local boundary conditions, where finite-difference discretizations ensure accurate approximations for infinite-dimensional systems. The results demonstrate how weaker topologies enhance analytical flexibility, stability and consistency, closing the gap between abstract operator theory and applications of partial differential equations (PDEs).