Some Aspects of Euclidean Spaces in Analysis and Linear Algebra

Abstract

The thesis comprise of six chapters. The chapter one is introductory. The chapter two is newlinedevoted to the Perfect sets. There is a problem Is there a nonempty perfect set in and#8477; newlinewhich contains no rational number? The answer to this question is yes and supporting newlineexamples are given in [24] and[30]. In this chapter we have fully illustrated a newlineconstruction of nonempty perfect subset of and#8477; containing no rational number such that newlineany graduate student in mathematics can easily be understand. Also we obtain infinite newlinesets which are disjoint bounded sets and unbounded sets. Moreover all of these sets are newlinemeasurable, with measure zero. The chapter three deals with Bolzano-Weierstrass newlineTheorem and Completeness of and#8477;and#1051616;. Using the result every sequence of real numbers newlinehas a monotone subsequence we prove: If a bounded sequence has unequal limit newlinesuperior and limit inferior, then it has at least two monotone (convergent) newlinesubsequences whose range sets are disjoint; (B. W. theorem) Every bounded sequence newlinehas a convergent subsequence, and and#8477; is a complete space under usual metric on and#8477;. newlineFrom these we obtain easy proofs of (B. W. theorem): Every bounded sequence in a newlineEuclidean space has a convergent subsequence, and every Euclidean space is complete. newlineFourth chapter is on Transfinite Cardinal Numbers. For transfinite cardinal number and#945;, newlineusing Zorn s lemma we have given a simple proof which is understandable to newlineundergraduate students, of the result and#945; + and#945; = and#945;and#945; = and#945;, that is, idempotency for addition newlineand multiplication. Moreover for a cardinal number and#61538; with 2 and#8804; and#61538; lt and#945; we obtain easily and#945; newline+ and#61538; = and#945;and#61538; = and#945;, and#945;and#61538; = and#945;and#945; = 2and#945;, and#945;and#61538; lt and#61538;and#945;. Using these results we get many results directly as newlineand#8501;0 + and#8705; = and#8501;0and#8705; = and#8705; + and#8705; = and#8705;and#8705; = and#8705;, and#8501;and#1051444; newlineand#8501;and#1051692; = and#8705; = and#8705;and#8501;and#1051692; = 2and#8501;and#1051692; , and#8705;and#8705; = and#8501;and#1051444; newlineand#8705; = 2and#8705; where and#8501;0 = card and#8469;, and#8705; newline= card and#8477;. Fifth chapter is devoted to Euclidean Spaces and Invariant Subspaces. For newlinea Euclidean space and#119881;n of odd dimension n, every linear operator T on and#119881;and#1051617;, the space and#119881;and#1051617; newlinehas a one and#8722; dimensional T and#8722; invariant subspace. If T has and#119899; distinct real eigen values newlinethen and#119881;and#1051617; has T-invar