Table of Contents
Table of Contents
"Introduction to Functional Analysis"
INTRODUCTION
FUNCTIONAL ANALYSIS
TRANSFORM
DISTRIBUTIONS
"Introduction to Functional Analysis"
"Introduction to Functional Analysis"
SIMONE MALACRIDA
In this book, aspects of functional analysis are presented with respect to:
Banach, Hilbert and Lebesgue spaces
measure according to Lebesgue and Lebesgue integral
operator view
discrete and continuous transforms
distributions and Sobolev spaces
––––––––
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
––––––––
INTRODUCTION
––––––––
I – FUNCTIONAL ANALYSIS
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 – TRANSFORM
Introduction and definitions
Fourier integral transform
Laplace integral transform
Other integral transforms
Discreet transforms
––––––––
III - DISTRIBUTIONS
Introduction and definitions
Operations
Sobolev spaces
INTRODUCTION
INTRODUCTION
Functional analysis is a branch of mathematics that is complementary to the more famous mathematical analysis.
As such, it intervenes in many aspects and in various results necessary for the resolution of mathematical and physical problems of various kinds.
Functional analysis starts from a rigorous definition of function spaces and from the study of the properties of these spaces, to then define increasingly complex operations.
With these formalisms it is possible to define transforms and distributions, two powerful methods for solving differential equations and analytic problems otherwise not known in their possible applications.
The knowledge required of the reader to understand this handbook is certainly university-level, given that, generally, the topics presented are carried out in advanced Mathematical Analysis courses (mathematical analysis 2 and mathematical analysis 3).
I
FUNCTIONAL ANALYSIS
FUNCTIONAL ANALYSIS
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 inequality holds:
The following function of n-dimensional R and C is called p-norm:
Norm 1 is the simple sum of the absolute values, norm 2 is the so-called Euclidean norm:
The infinite norm is instead defined as follows:
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 19.04.2023
ISBN: 978-3-7554-3943-1
Alle Rechte vorbehalten