Cone Functional Analysis And The Mazur-Ulam Theorem

There are five chapters in the paper.In Chapter1, we mainly introduce some preliminary knowledge. We also list some marks of the concept which is used in the paper.In Chapter2, we study the nature of the cone and cone metric space. And then show the concept of cone normed linear space and boundedness linear operator between two spaces. Finally, we research some basic properties of cone normed linear space, and obtain the famous Riesz lemma of the classical normed linear space.In Chapter3, by using the conclusion in the previous chapter, we further s-tudy theory of boundedness linear operator between two cone normed linear spaces. Through the concept of dual space, we give the concept of Banach adjoint operator, annihilator, quotient space and so on. Eventually, we obtain some famous theorem of classical normed linear space. Such as Hahn-Banach theorem, Baire outline theorem, principle of uniform boundness, open mapping theorem, Banach inverse operator theorem, closed graph theorem, closed range theorem and so on.In Chapter4, we point out that there are some mistakes in the proof of two theorems, which are the Concave Programming Theorem and the Conjugate Theo-rem, of convex analysis problem. firstly, we will be the concept of terminal subset of convex set generalized to the endset of general set. Then, we use it to prove the concave programming theorem. Secondly, we give a lemma in locally convex space and use it to prove the Conjugate Theorem of conjugate function.In Chapter5, though the proof of Mazur-Ulam theorem, we give a simplified proof of the theorem. Then put the theorem for promotion.At last, we summarize work of this paper, and Put forward some unsolved problems.