NON-LINEAR TRANSFORMATIONS AND WEAKLY SEMINORMED VECTOR SPACES: STRUCTURAL CHARACTERIZATION, STABILITYCONDITIONS, AND FUNCTIONAL IMPLICATIONS

Weakly seminormed vector spaces provide a flexible analytical setting for situations in which the classical assumptions of normed and seminormed spaces are only partially valid.These spaces support weaker forms of convergence and continuity, making them particularly relevant in contexts shaped by nonlinear behaviour and irregular transformation patterns.The present study develops a structured characterization of weakly seminormed vector spaces and examines how their properties respond to different classes of non-linear transformations.The analysis focuses on stability, deformation, and preservation of seminorm structures, with special attention to quasi-linear, sub-additive, and locally Lipschitz mappings.By making use of quotient-space representations and weak topological arguments, the study identifies conditions under which weak seminorms remain stable and conditions under which controlled deformation occurs without loss of analytical meaning.At the same time, the paper also highlights situations in which nonlinear behaviour leads to structural breakdown, illustrating the limits within which weak seminorm frameworks remain valid.Examples and counterexamples are used to clarify the theoretical results and to demonstrate how different transformation classes influence the underlying structure of the space.Overall, the findings contribute to a deeper understanding of the interaction between weak topological geometry and non-linear functional behaviour, and they offer conceptual foundations for further work in approximation theory, optimization models, and generalized operator analysis.