Infinite matrix method, ($\boldsymbol~{op}$) type space, invariance and demi-linear analysis

This paper summarizes the research work on functional space theory, including the infinite matrix method, characterization of $({op})$ type space,invariance, nonlinear open mapping theorem, nonlinear closed graph theorem, demi-linear equicontinuity theorem, demi-linear duality theory, anddemi-linear generalized function theory.