OICC Press Mathematical Analysis and its Contemporary Applications 7 4 2025 11 10 Norm-attainment in locally convex spaces: Weak-* topology, inductive limits, and reflexivity 10.30495/maca.2025.2057641.1134 EN Evans N. Mogoi Jaramogi Oginga Odinga University of Science and Technology, Kenya Evans N. Mogi Jaramogi Oginga Odinga University of Science and Technology, Kenya Priscah Moraa Ohuru Jaramogi Oginga Odinga University of Science and Technology, Kenya Journal Article 2025 11 10 We characterize norm-attaining functionals in locally convex spaces (LCS), with particular focus on three fundamental aspects: the weak-* (weak-star) topology in dual spaces, inductive limits (including LF-spaces and DF-spaces), and reflexivity conditions. Our main results establish that (1) norm-attainment in the weak-* dual coincides precisely with the canonical embedding X into X**; (2) strict inductive limits (such as D(R)) permit non-attaining functionals, whereas Montel spaces ensure universal attainment; and (3) both barrelledness and reflexivity conditions recover norm-attainment through weak-* continuity. This work extends classical Banach space techniques to general LCS settings, revealing the crucial interplay between compactness properties and approximation methods in determining norm-attainment behavior.