Compactness and Finite-Dimensional Structure of Intuitionistic Algebra Fuzzy Normed Spaces

Abstract:This study develops compactness and finite-dimensionality results for intuitionistic algebra fuzzy normed spaces. It begins by presenting the fundamental concepts of intuitionistic algebra fuzzy norms, convergence, Cauchy sequences, boundedness, fuzzy balls, closed sets, and limit points. Compactness is then defined through intuitionistic fuzzy open covers and finite subcovers. The authors establish relationships among compactness, total boundedness, completeness, sequential compactness, convergent subsequences, and limit points. They prove that every intuitionistically fuzzy compact space is both totally bounded and complete and, conversely, that total boundedness combined with completeness implies compactness. The second part addresses finite-dimensional spaces. An estimate for linear combinations of linearly independent vectors is established and used to demonstrate that every finite-dimensional subspace is complete. The study also proves that any two intuitionistic algebra fuzzy norms on a finite-dimensional vector space are equivalent. A generalized Heine–Borel theorem follows, according to which closed and bounded subsets of finite-dimensional intuitionistic algebra fuzzy normed spaces are compact. These findings extend central results of classical functional analysis to an intuitionistic fuzzy setting.