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
––––––––
INTRODUCTION
––––––––
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
––––––––
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
––––––––
Functional analysis is that part of mathematical analysis that deals with the study of spaces of functions.
––––––––
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:
Given two normed spaces (we will describe their characteristics shortly) one of which is included in the other, if the inclusion function is continuous then we say that the first is continuously immersed in the second.
Furthermore, if any bounded set in the first space is precompact in the other space (that is, any subsequence in that bounded set has a subsequence that is Cauchy in the reference norm), then the first space is compactly embedded in the second.
––––––––
A particularly important result of mathematical analysis in functional analysis is the Ascoli-Arzelà theorem.
A sequence of uniformly bounded continuous functions is equicontinuous if:
The theorem states that an equicontinuous and uniformly bounded sequence admits a uniformly convergent subsequence.
––––––––
Norms and regulated spaces
––––––––
A pseudometric space is a space that has the same characteristics as a metric space except that the distance is different from zero for each pair of distinct points.
say ultrametric space , a space in which the triangular inequality takes this form:
We define the norm on a real or complex vector space as a homogeneous function, positive definite and such for which the triangular
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 23.04.2023
ISBN: 978-3-7554-4003-1
Alle Rechte vorbehalten