IN THIS DISSERTATION WE STUDY THE NON-ARCHIMEDEAN WEIGHTED SPACES OF CONTINUOUS FUNCTIONS CV(X,E), WHERE X IS A HAUSDORFF TOPOLOGICAL SPACE, V A NACHBIN FAMILY ON X AND E A NON-ARCHIMEDEAN LOCALLY CONVEX SPACE. THE NOTIONS OF THE POLAR NON-ARCHIMEDEAN LOCALLY CONVEX SPACE E AND THE ZERODIMENSIONAL TOPOLOGICAL SPACE X ARE VERY IMPORTANT FOR THE DEVELOPMENT OF OUR THEORY. IN SEVERAL CASES OUR RESULTS ARE ANALOGOUS TO THE ONES OF THE CLASSICAL CASE, ALTHOUGH THE PROVING METHODS ARE OF TEN OFTEN COMPLETELY DIFFERENT. WE GIVE SUFFICIENT CONDITIONS FOR CV(X,E) TO BE COMPLETE, OF COUNTABLE TYPE, QUASINORMABLE, (DF) AND Σ-LOCALLY TOPOLOGICAL RESPECTIVELY. WE ALSO PROVE THAT ON CERTAIN CONDITIONS, THE TOPOLOGICAL DUAL OF THE SUBSPACE CVO(X,E) OF CV(X,E) IS ALGEBRAICALLY ISOMORPHIC TO A FINITELY ADDITIVE MEASURE SPACE. WE GIVE SOME SUFFICIENT CONDITIONS SO AS THE COMPACTOID SUBSETS OF CVO(X,E) TO BE DESCRIBED BY AN ARZELA-ASCOLI THEOREM. AT LAST, WE STUDY THE Π-TENSOR PRODUCT CVO(X)#ΠCUO(Y,E) AND WE STATE A SUFFICIENT CONDITION SO AS CVO(X)#ΠCUO(Y,E) BE TOPOLOGICALLY ISOMORPHIC TO A SUBSPACE OF A NON-ARCHIMEDEAN WEIGHTED SPACE OF CONTINUOUS FUNCTIONS.
University of Ioannina
