Isomorphisms generated by fundamental and total sets

A biorthogonal system {x?, /,•} ,■<=/ in a linear topological duality (X, X*) is a Generalized Basis (G-Basis) if {/,■} is total over X; a Dual Generalized Basis (DG-Basis) if {xi} is fundamental in X. Arsove and Edwards [l] proved that the existence of similar G-Bases in Frechet spaces X and Y implies that X and F are isomorphic (linearly homeomorphic). Davis [2] noted that if {xi, /,} is a DGBasis for X, then {/,-, x,} is a G-Basis for X* if X* is endowed with any topology stronger than w(X*, X). This observation and the isomorphism theorem of Arsove and Edwards allowed him to prove that the existence of *similar DG-Bases in Banach spaces X and Y implies that X and Y are isomorphic. The definitions of similarity and *similarity can be extended to structures more general than Gand DG-Bases (Definitions 1 and 2). Similarity implies the existence of an algebraic isomorphism with closed graph between the spaces (Theorem 1), and thus the ArsoveEdwards result is extended to fully complete barreled spaces. The assumption of *similarity yields an isomorphism theorem for a class of spaces which includes Frechet spaces (Theorem 2). The notation and terminology of [5] is used throughout.