tailieunhanh - Real Analysis with Economic Applications - Chapter J

Chapter J Normed Linear Spaces This chapter introduces a very important subclass of metric linear spaces, namely, the class of normed linear spaces. We begin with an informal discussion that motivates the investigation of such spaces. We then formalize parts of that discussion, introduce Banach spaces | Chapter J Normed Linear Spaces This chapter introduces a very important subclass of metric linear spaces namely the class of normed linear spaces. We begin with an informal discussion that motivates the investigation of such spaces. We then formalize parts of that discussion introduce Banach spaces and go through a number of examples and preliminary results. The first hints of how productive mathematical analysis can be within the context of normed linear spaces are found in our final excursion to fixed point theory. Here we prove the fixed point theorems of Glicksberg Fan Krasnoselskii and Schauder and provide a few applications to game theory and functional equations. We then turn to the basic theory of continuous linear functionals defined on normed linear spaces and sketch an introduction to classical linear functional analysis. Our treatment is guided by geometric considerations for the most part and dovetails with that of Chapter G. In particular we carry our earlier work on the Hahn-Banach type extension and separation theorems to the realm of normed linear spaces and talk about a few fundamental results of infinite dimensional convex analysis such as the Extreme Point Theorem Krein-Milman Theorem etc. In this chapter we also bring to the conclusion our work on the classification of the differences between the finite and infinite dimensional linear spaces. Finally in order to give at least a glimpse of the powerful Banach space methods we establish here the famous Uniform Boundedness Principle as a corollary of our earlier geometric findings and go through some of its applications. The present treatment of normed linear spaces is roughly at the same level with that of the classic real analysis texts by Kolmogorov and Fomin 1970 and Roy-den 1994 . For a more detailed introduction to Banach space theory we should recommend Kreyzig 1978 Maddox 1988 and or the first chapter of Megginson 1998 .1 1My coverage of linear functional analysis here is directed towards