On the Closed Graph and Open Mapping Theorems

Abstract

<p> The closed graph and open mapping theorems are two of the deeper results in the theory of locally convex spaces. They are very rich in their applications in functional analysis. This thesis contains some extensions of these theorems in locally convex spaces. We begin with a study of α-spaces, and γ-spaces, which leads us naturally to a study of δ-spaces. On these spaces, we prove closed graph and open mapping theorems. Similar theorems are also proved for certain classes of Br ('&)-spaces. In particular, a closed graph theorem for B(m)-spaces enables us to characterise certain classes of B( &. y)-spaces. A consideration of countability conditions in locally convex spaces enables us to prove open mapping theorems in Br (J )-spaces. These theorems are then used to relate boundedness of linear mappings and their graphs. </p>