LOCALLY CONVEX SPACES WITH THE B(TAU)-PROPERTY
Abstract
A locally convex topological vector space E is said to have the B( )-property or, in short, to be a B( )-space if, for each t-space F, a linear and continuous mapping f of E onto F is open. If, in addition, f is one-to-one, then E is said to be a Br( )-space. Besides a characterization of B( )-spaces and other results, the following theorems have been proved. Theorem: Let E be a metrizable locally convex space. Then the dual E' of E with any locally convex topology finer than (E',E) and coarser than (E',E) is a B( )-space. This theorem provides many examples of B( )-spaces which are not B-complete. However, it is easy to see that a B-complete space is a B( )-space. A very general closed graph theorem has been proved from which result the well-known closed graph the rem for B-complete spaces and also the following theorem: Let F be both a t-space and a Br( )space. Let f be a linear mapping of any locally convex space E into F with the closed graph. If f is almost continuous, then f is continuous. An example is given to show that the above theorem is false if F is not a t-space. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jul 01, 1961
- Accession Number
- AD0261482
Entities
People
- Taqdir Husain
Organizations
- Syracuse University