Mathematical Analysis

1. Front Matter

   Pages i-xix

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2. Linear Algebra

   1. Front Matter

      Pages 1-1

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   2. Vectors, Matrices and Linear Systems

      Pages 3-39

   3. Vector Spaces and Linear Maps

      Pages 41-78

   4. Euclidean and Hermitian Spaces

      Pages 79-110

   5. Self-Adjoint Operators

      Pages 111-145

3. Metrics and Topology

   1. Front Matter

      Pages 147-147

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   2. Metric Spaces and Continuous Functions

      Pages 149-195

   3. Compactness and Connectedness

      Pages 197-217

   4. Curves

      Pages 219-247

   5. Some Topics from the Topology of ℝn

      Pages 249-282

4. Continuity in Infinite-Dimensional Spaces

   1. Front Matter

      Pages 283-283

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   2. Spaces of Continuous Functions, Banach Spaces and Abstract Equations

      Pages 285-350

   3. Hilbert Spaces, Dirichlet’s Principle and Linear Compact Operators

      Pages 351-394

   4. Some Applications

      Pages 395-453

5. Back Matter

   Pages 455-466

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One of the fundamental ideas of mathematical analysis is the notion of a function; we use it to describe and study relationships among variable quantities in a system and transformations of a system. We have already discussed real functions of one real variable and a few examples of functions of several variables but there are many more examples of functions that the real world, physics, natural and social sciences, and mathematics have to offer: (a) not only do we associate numbers and points to points, but we as- ciate numbers or vectors to vectors, (b) in the calculus of variations and in mechanics one associates an - ergy or action to each curve y(t) connecting two points (a, y(a)) and (b,y(b)): b Lea ~(y) - / 9 F(t, y(t), y' (t))dt t. J a in terms of the so-called Lagrangian F(t, y, p), (c) in the theory of integral equations one maps a function into a new function b /1, d-r / o. J a by means of a kernel K(s, T), (d) in the theory of differential equations one considers transformations of a function x(t) into the new function t t f f( a where f(s, y) is given. 1 in M. Giaquinta, G. Modica, Mathematical Analysis. Functions of One Va- able, Birkh~user, Boston, 2003, which we shall refer to as [GM1] and in M. G- quinta, G. Modica, Mathematical Analysis. Approximation and Discrete Processes, Birkhs Boston, 2004, which we shall refer to as [GM2].

• calculus
• compactness
• differential equation
• self-adjoint operators
• spectral theorem
• topology of metric spaces
• vector spaces
• ordinary differential equations

From the reviews:

"This book is suitable as a text for graduate students. Photographs of Banach, Fréchet, Hausdorff, Hilbert and some others mathematicians are imprinted in order to involve [the reader] in the work of mathematicians."—Zentralblatt MATH

"This volume is an English translation and revised edition of a former Italian version published in 2000. … This nice book is recommended to advanced undergraduate and graduate students. It can serve also as a valuable reference for researchers in mathematics, physics, and engineering." (L. Kérchy, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

“The book ‘M. Giaquinta, G. Modica: Mathematical Analysis. Linear and Metric Structures and Continuity’ is a lovely book which should be in the bookcase of every expert in mathematical analysis.” (Dagmar Medková, Mathematica Bohemica, Issue 2, 2010)

“This book offers a self-contained introduction to certain central topics of functional analysis and topology for advanced undergraduate and graduate students. … the clear and self-contained style recommend the book for self-study, offering a quick introduction to a number of central notions of functional analysis and topology. A large number of exercises and historical remarks add to the pleasant overall impression the book leaves.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 157 (2), June, 2009)

• Dipartimento di Matematica, Scuola Normale Superiore, Pisa, Italy

  Mariano Giaquinta

• Dipartimento di Matematica Applicata, Università degli Studi di Firenze, Firenze, Italy

  Giuseppe Modica

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