Some Geometric Properties Of Banach Spaces

This paper is divided into two parts. The first part discusses the geometric properties in Banach spaces, including (K) convexity, (K)smoothness, (K) differentiability, modulus, frame, Riesz basis and Bessel sequence, etc. In chapter1st, the author summarizes different kinds of definitions of (K) convexity, (K) smoothness, anduses vector norm to define K Gateaux differentiability and K Fréchet differentiability, thendiscusses the relationships among (K) convexity, (K) smoothness, and (K) differentiability,respectively, discusses the dual relationship between (K) convexity and (K) smoothness, and givessome equivalent conclusions about (K) smoothness(theorem1.2.19, 1.2.20). The author gives somenecessary and sufficient conditions of smoothness, extreme smoothness, uniform smoothness,very smoothness, Fréchet differentiability, and so on. Chapter 2nd discusses modulus to describe(K) convexity, (K) smoothness, near (K) convexity, near (K) smoothness. In chapter 3rd,according to frames, Riesz basis and Bessel sequence in Banach Spaces and Hilbert, the authordiscusses the properties of p-frame, exact p-frame; p-Riesz basis , p-Bessel sequence and dualp-frame in Banach spaces (§2). Then introduces N-frame and M-Riesz basis given by Zhu. The second part discusses the application of geometric properties in Banach spaces to Ba spaces,Da spaces and Fa spaces. Chapter 1st studies many properties of Ba spaces as Banach Spaces(theorem 1.2.7~1.2.22), then gives inerpolation of Ba spaces which had been discussed by Liuwho studies them based on Chen and Meng. In chapter 2nd and 3rd Da spaces and Fa spaces aredefined as Ba spaces, respectively, and discusses their properties too. Da spaces are generated by aclass of metric spaces, and Fa spaces are generated by a class of linearly quasi-normed spaces.Chapter fourth gives Fourier transform (theorem 4.1.1) under two limited conditions in Ba spaceswhich is formed by Lp(R1) and get Titchmarsh inequality.