Exercises of Functional Analysis E-Book

Simone Malacrida

• Herausgeber: BookRix
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2023

Beschreibung

In this book, exercises are carried out regarding the following mathematical topics: Banach and Hilbert spaces operations in vector spaces Lebesgue measure and integral. Initial theoretical hints are also presented to make the conduct of the exercises understandable.

Leseprobe

Simone Malacrida

Exercises of Functional Analysis

BookRix GmbH & Co. KG81371 Munich

Table of Contents

Table of Contents

“Exercises of Functional Analysis”

INTRODUCTION

THEORETICAL OUTLINE

EXERCISES

“Exercises of Functional Analysis”

“Exercises of Functional Analysis”

SIMONE MALACRIDA

In this book, exercises are carried out regarding the following mathematical topics:

Banach and Hilbert spaces

operations in vector spaces

Lebesgue measure and integral.

Initial theoretical hints are also presented to make the conduct of the exercises understandable.

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

ANALYTICAL INDEX

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INTRODUCTION

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I – THEORETICAL OUTLINE

Introduction and definitions

Norms and regulated spaces

Hilbert spaces

Lebesgue measure and Lebesgue integral

Lebesgue spaces

Other results of functional analysis and operative vision

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II – EXERCISES

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Exercise 12

Exercise 13

Exercise 14

Exercise 15

Exercise 16

Exercise 17

Exercise 18

Exercise 19

Exercise 20

Exercise 21

Exercise 22

INTRODUCTION

INTRODUCTION

In this workbook, some examples of calculations related to functional analysis are carried out.

Furthermore, the main theorems used in this area of mathematics are presented.

Functional analysis completes and enriches the study of mathematical analysis by proposing new solutions and new fields of application.

In fact, without the typical setting of functional analysis, the integral transforms and distributions would not be determined.

In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.

What is exposed in this workbook is generally addressed in advanced mathematical analysis courses (analysis 3).

I

THEORETICAL OUTLINE

THEORETICAL OUTLINE

Introduction and definitions

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Functional analysis is that part of mathematical analysis that deals with the study of spaces of functions.

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We define embedding as a relationship between two mathematical structures such that one contains a subset of the other and retains its properties.

Essentially, immersion extends the concept of set inclusion to functional analysis.

A mathematical structure is immersed in another if there is an injective function such that the image of the first structure according to the function preserves all, or even only part, of the mathematical structures.

Set inclusion is an immersion that is called canonical.

A topological embedding between two topological spaces is an embedding if it is a homeomorphism.

An embedding between metric spaces is a relation which maintains the concept of distance, up to a bias factor.

Given a topological space and two subsets V and W of it, V is said to be compactly embedded in W if the closure of V is compact and if: