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Title:	Exercises on Normed Vector Spaces
Authors:	Rakia, Ahmed-Yahia.
Issue Date:	2-Jul-2025
Publisher:	University Center of Abdalhafid Boussouf - MILA
Description:	The understanding of results and notions for a student in mathematics requires solving exercises. The exercises are also meant to test the reader’s understanding of the text material, and to enhance the skill in doing calculations. The goal of this collection of exercises is to help the student understand much better the basic facts which are usually presented in an introductory course in normed vector spaces, by providing various exercises, from different topics, from simple ones to, perhaps, more difficult ones, in order to help him to understand the richness of ideas and techniques which Banach spaces in general and Hilbert ones in particular offer. These exercises are related to a course given to students in the 5th semester of mathematics study. We will practice on two chapters in normed vector spaces, wich are Banach spaces in general and Hilbert spaces as a particular case of them. For the first, we start by familiarizing the student with the norm through elementary calculations, and various spaces, while showing the topological aspect of normed spaces. Then Cauchy sequences are introduced in order to clarify the concept of completeness, and subsequently practice on Banach spaces. Next comes the continuous linear applications which define another normed vector space, among others the dual space. The second one, we first practice the scalar product, by memorizing basic results, then Hilbert spaces with the Riesz Theorem , after which we move on to the projection theorem and finish with Fourier series by training on particular trigonometric systems. we first set the exercises, then give the solution with many details, and finaly propose some non solved exercises left to the student, in order to push him to work and investigate. I hope that some of the exercises herein can be of some help to the teacher of Normed vector spaces as seminar tools; and to anyone who is interested in seeing some applications of normed vector spaces.
URI:	http://dspace.centre-univ-mila.dz/jspui/handle/123456789/4105
Appears in Collections:	Mathematics

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