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Mathematical Analysis is a very broad discipline, which seems to have no boundary. It is the only discipline to give certainty to the human mind, but it demands from those who wish to use it an absolute logical rigor.
The Mathematical Analysis meets needs of mathematicians, of scientists and of engineers who are working at frontiers of their fields. They often have to use the modern Mathematical Analysis to formulate their particular problem. They often have to also modify some result of the modern Mathematical Analysis to be able to solve their particular problem.
This book is an introduction to modern Mathematical Analysis, written for individual study. It needs, as unique prerequisite for reading the book, the classical real Mathematical Analysis, i.e., the study of the real number field R and the study (limits, continuity, differentiation, integration) of the functions that map R or Rn into R or Rn .
The rigor of exposure, the modern and sufficiently general conception make this book as well a useful basis for the challenging study of the modern Mathematical Analysis.


Contents
Chapter 1Set Theory…………………….……1
1.1 Basics of Sets…….……..……………………..1
1.1.1 Logical Preliminaries………………………1
1.1.2 Basic notions about Sets……………………...2
1.1.3 Cartesian product of Sets……………………..13
1.2 Relation …………………………………………...14
1.2.1 Equivalence relation.…………………………15
1.2.2 Order relation.………………………………..17
1.3 Functions …………………………………….22
1.3.1 Basics of Functions...………………………...22
1.3.2 Sequence...………………………...................26
1.3.3 Composite function.………………………….27
1.3.4 Monotonic function…………………………..28
1.3.5 Other basic properties of functions…………...29

Chapter 2 Topological Spaces…………….34
2.1 Topological Spaces.……………..……………34
2.1.1 Topology. Open set.………………………….34
2.1.2 Neighborhood. Accumulation point...……….38
2.1.3 Closed set. Closure..……………………….…41
2.1.4 Compact set………….……………………….47
2.2 Metric Spaces……………………………………..50
2.2.1 Metrics………………………………………..50
2.2.2 Contraction …………………………………….60

Chapter 3Topological Vector Spaces.…… 63
3.1 Topological Vector Spaces…………………….63
3.1.1 Vector space……………………………………63
3.1.2 Topological vector space………………………67
3.1.3 Normed space…………………………….........79
3.1.4 Neighborhood systems………………………...94
3.2 Banach spaces…………………………………115
3.2.1 Banach space…………………………………..115
3.2.2 Reflective space………………………………..121
3.2.3 Quotient space…………………………………125
3.2.4 Gateaux differentiability………………………131
3.2.5 Minimum of a functional……………………...138
3.3 Hilbert spaces…………………………………143
3.3.1 Hilbert space…………………………………..143
3.3.2 Projection……………………………………...146
3.3.3 Sesquilinear form……………………………...154
3.3.4 Immersion……………………………………..168
3.3.5 Compact operators…………………………….181

BIBLIOGRAPHY
1. RUDIN W., Functional Analysis, Tata McGraw-Hill Publishing C., 1981
2. SCHWARTZ L., Analyse. Topologie générale et analyse fonctionnelle,Hermann, 1970
3. HEWITT E. – STROMBERG K., Real and Abstract Analysis, Springer, 1975
4. YOSIDA K., Functional Analysis, Springer, 1980
5. DUNFORD N. – SCHWARTZ J.T., Linear Operators, Interscience Publishers, 1976
6. MACERI A., Mathematical Analysis, Indep. publ., 2024
7.MACERI A., Theory of Elastcity, Springer, 2010.

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