Let D⊂ CN be a domain, H be a locally convex space with the topology defined by a sequence of Hilbert semi-norms. Denote H(D,H) the space of H-valued holomorphic functions on D and R(CN,H) the space of H-valued rational functions on CN. In this paper, we set up terminology of an admissible sequence W = {wm}m≥1 of weights for H(D, F) and study sufficient conditions on W to ensure convergence of wm(||rm-f||m2) (pointwise/ in capacity/ uniformly on compact subsets) to 0 on the whole domain D except for a pluripolar subset provided the pointwise (rapid) W-convergence of {rm-f}m≥1 to 0 occurs on a Borel non-pluripolar subset X for every f ∈ H(D,H) and {rm}m≥1 ⊂ R(CN,H), in both of cases X lies in D, and X lies in the boundary ∂D of a bounded domain D. Date: January 01, 2023. 2020 Mathematics Subject Classification. 41A20, 46G20, 31C10, 31C15