(untitled) (European Journal of Pure and Applied Mathematics, 2026)
We introduce Generalized Inner Structure Spaces (GISS), a class of Fréchet spaces equipped with a sesquilinear form \([·, ·]\) that generalizes Hilbert spaces by allowing indefinite or degenerate forms, unifying Krein spaces, semi-inner product spaces, Gelfand triples, and related structures. We develop a comprehensive theory, establishing a locally convex topology \(τ_p\) induced by seminorms \(p_x(y) = |[x, y]|\) (Section sec:topology), a Hahn-Banach-type separation theorem (Theorem thm:hahn-banach), and operator theory for self-adjoint operators with real spectra (Theorem thm:resolvent). A key result is the decomposition \(X = X_+ X_- X_0\) into positive, negative, and isotropic subspaces (Theorem thm:decomposition). Examples in sequence spaces, Sobolev spaces, Gelfand triples, and quantum state spaces illustrate GISS’s versatility (Section sec:examples). Applications include quantum field theory (e.g., Dirac operator quantization) and hyperbolic PDEs (Section sec:applications). Open problems address spectral decomposition and unbounded operators. GISS offers a robust framework for non-Hilbertian analysis in functional analysis and mathematical physics.