The Theory Of Menger Probabilistic Inner Product Spaces

The purpose of this paper is to research the probabilistic inner product spaces. With the tool of semi-inner product, Schwarz inequality, orthogonal projection theorem and Riesz representation theorem of probabilistic inner product spaces are established.This dissertation includes three parts:In Chapter 1, to prepare for the next part, we introduce the definition of probabilistic inner product spaces, the completeness and convergence of probabilistic inner product spaces. The definition of semi-inner product is. given. Then the characteristics of this kind of semi-inner product are discussed.In Chapter 2, with the tool of semi-inner product, Schwarz inequality and parallelogram formula of probabilistic inner product spaces are established in general distribution function set. Then the continuity of semi-norm and semi-inner product is discussed. We describe convergence by semi-norm. Then the convergence of the point sequence and completeness of probabilistic inner product spaces are discussed.In Chapter 3, we discussed the orthogonality of elements in the probabilistic inner product spaces. The definition of orthogonal decomposition and orthogonal projection are introduced. Minimum vector theorem and projection theorem are established. At last, the Riesz representation theorem of probabilistic inner product spaces is established by means of semi-inner product.