# Accession Number:

## AD0261482

# Title:

## LOCALLY CONVEX SPACES WITH THE B(TAU)-PROPERTY

# Descriptive Note:

# Corporate Author:

## SYRACUSE UNIV N Y

# Personal Author(s):

# Report Date:

## 1961-07-01

# Pagination or Media Count:

## 1.0

# 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