Some Geometric Properties Of Banach Space In The Sense Of Ellipsoid

Uniformly non-square, uniformly convex and strictly convex are importantgeometric properties of Banach space. They play very important roles inmathematics and they have important applications in all kinds of areas, such asprobability theory, approximation theory and so on. For a long time, geometricproperties of space are studied by using the unit ball. With further research, peopletry to substitute other geometric figures for unit ball, and to study other geometricproperties of geometry. Therefore, the geometric properties of Banach space in thesense of ellipsoid have theoretical significance and application value. The maincontents of this thesis are as follows:This thesis shows necessary and sufficient conditions for Banach spaces whichare uniformly convex, strictly convex and uniformly non-square in the sense ofellipsoid respectively. Convexity and non-square property in the sense of ellipsoidare discussed by using convex module. The main contents of this dissertation aredivided into three parts:The first part reviews the development process of Banach space theory andintroduces the development of non-square property and convex property briefly.The second part discusses some geometric properties of Banach space in thesense of ellipsoid and gives the relevant proof about necessary and sufficientconditions for Banach spaces which are uniformly convex, strictly convex anduniformly non-square property in the sense of ellipsoid respectively, and discussesthe relevance of non-square property in general meaning and in the sense ofellipsoid.In the third part, the necessary and sufficient conditions for normed spacewhich are convexity and non-square property in the sense of ellipsoid is discussedby using convex module.