Introduction to Functional Analysis

By 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

• Language: English
• Publisher: Simone Malacrida
• Release date: Dec 22, 2022
• ISBN: 9798215463147

Simone Malacrida

Simone Malacrida (1977) Ha lavorato nel settore della ricerca (ottica e nanotecnologie) e, in seguito, in quello industriale-impiantistico, in particolare nel Power, nell'Oil&Gas e nelle infrastrutture. E' interessato a problematiche finanziarie ed energetiche. Ha pubblicato un primo ciclo di 21 libri principali (10 divulgativi e didattici e 11 romanzi) + 91 manuali didattici derivati. Un secondo ciclo, sempre di 21 libri, è in corso di elaborazione e sviluppo.

Book preview

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

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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 – 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

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II – TRANSFORM

Introduction and definitions

Fourier integral transform

Laplace integral transform

Other integral transforms

Discreet transforms

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III - DISTRIBUTIONS

Introduction and definitions

Operations

Sobolev spaces

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

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