BARTLE Integral On Locally Convex Space

This thesis is mainly based on papers《A general bilinear vector integral》of R.G.Bartle and《A New Limit Theorem of Bartle Integration》of Zhao Huanguang. We extended several conclusions of the Bartle integral on Banach spaces to separated locally convex spaces. In view of the representation theorem in locally convex operators space, we studied the sequential completeness and P^**—completeness of bounded vector measure space ba(F, X).From this we knew that the sequential completeness of locally convex spaces has the "lifting character" . We introduced the concept for a F-integrable function and the definition of a new Bartle integration for countably additive vector measures .Then we got some basic properties for them. In the final part of this thesis we gave two limit theorems for Bartle integration on locally convex space and some corollaries for example,the dominated convergence theorem,the bounded convergence theorem.