Generalized Inner Structure Spaces: Indefinite Sesquilinear Forms in Fréchet Spaces and Their Applications

Abstract

We introduce \emph{Generalized Inner Structure Spaces} (GISS), a class of Fréchet spaces equipped with a sesquilinear form \([\cdot, \cdot]\) 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 \(\tau_p\) induced by seminorms \(p_x(y) = |[x, y]|\) (Section \ref{sec:topology}), a Hahn-Banach-type separation theorem (Theorem \ref{thm:hahn-banach}), and operator theory for self-adjoint operators with real spectra (Theorem \ref{thm:resolvent}). A key result is the decomposition \(X = X_+ \oplus X_- \oplus X_0\) into positive, negative, and isotropic subspaces (Theorem \ref{thm:decomposition}). Examples in sequence spaces, Sobolev spaces, Gelfand triples, and quantum state spaces illustrate GISS’s versatility (Section \ref{sec:examples}). Applications include quantum field theory (e.g., Dirac operator quantization) and hyperbolic PDEs (Section \ref{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.