Partial inner product spaces and semi-inner product spaces

Abstract A comparison is made between the two objects mentioned in the title. Connections between them are threefold: (i) both are particular instances of dual pairs of locally convex spaces; (ii) many partial inner product spaces consist of chains or lattices of semi-inner product spaces; (iii) the basic structure behind both of them is that of Galois connections. A number of common open problems are described.

[2]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[3]  Laurent Schwartz,et al.  Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (Noyaux reproduisants) , 1964 .

[4]  Günter Pickert.  Bemerkungen überGalois-Verbindungen , 1952 .

[5]  J. Diestel.  Geometry of Banach Spaces: Selected Topics , 1975 .

[6]  E. J. McShane.  Linear functionals on certain Banach spaces , 1950 .

[7]  J. Antoine.  Partial inner product spaces. III. Compatibility relations revisited , 1980 .

[8]  Jean-Pierre Antoine.  Partial inner product spaces: Another approach to unbounded operators , 1978 .

[9]  A. Grossmann,et al.  A class of explicitly soluble, local, many‐center Hamiltonians for one‐particle quantum mechanics in two and three dimensions. I , 1980 .

[10]  J. Giles.  Classes of semi-inner-product spaces , 1967 .

[11]  A. Grossmann,et al.  Partial inner product spaces. I. General properties , 1976 .

[12]  F. Bonsall,et al.  Numerical Ranges II , 1973 .

[13]  Yu. I. Petunin,et al.  SCALES OF BANACH SPACES , 1966 .

[14]  G. Lumer.  SEMI-INNER-PRODUCT SPACES , 1961 .

[15]  J. Duncan,et al.  Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras: Hermitian elements of a complex unital Banach algebra , 1971 .

[16]  h.c. Gottfried Köthe.  Topological Vector Spaces II , 1979 .

[17]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[18]  F. L. Bauer.  On the field of values subordinate to a norm , 1962 .

[19]  R. Phillips,et al.  Dissipative hyperbolic systems , 1957 .