Some Problems On Best Apporoximation In Infinite Direct Sums

In chapter 1 we mainly discuss the character theorems of best approx-imation in substitution spaces.Following we give a sufficient condition of the existence by an analogous metho d.At last,on the presupposition of sun set,we discuss the uniqueness of the best approximation,and present the necessary and sufficient conditions for the unique existence of the best approximation.Also,we give some examples to illustrate the principle. In chapter 2,on one hand we extend the conclusions of the chapters 1 into PBBs.on the other hand we explain greatly to demonstrate that the main results of this paper are the abstractness and extension of the simultaneous approximation recently.