Search results for "Equicontinuity"

On Certain Metrizable Locally Convex Spaces

1986

Publisher Summary This chapter discusses on certain metrizable locally convex spaces. The linear spaces used are defined over the field IK of real or complex numbers. The word "space" will mean "Hausdorff locally convex space". This chapter presents a proposition which states if U be a neighborhood of the origin in a space E. If A is a barrel in E which is not a neighborhood of the origin and F is a closed subspace of finite codimension in E’ [σ(E’,E)], then U° ∩ F does not contain A° ∩ F. Suppose that U° ∩ F contain A° ∩ F. Then A° ∩ F is equicontinuous hence W is also equicontinuous. Since W° is contained in A, it follows that A is a neighborhood of the origin, a contradiction.