A First Course in Functional Analysis

1. Auflage 2027
ISBN: 978-1-041-30984-0
Verlag: Taylor & Francis Ltd

Buch, Englisch, 368 Seiten, Format (B × H): 178 mm x 254 mm

A First Course in Functional Analysis offers a rigorous and self-contained introduction to the basic language, methods, and theorems of functional analysis. Written for advanced undergraduate students, beginning graduate students, and instructors, the book develops the subject without assuming measure theory or the Lebesgue integral as prerequisites. This choice allows the essential ideas of the theory to be presented through vector spaces, normed spaces, Banach spaces, Hilbert spaces, and linear operators, while maintaining a gradual and conceptually coherent path through the material.

The book begins with algebraic foundations, including fields, valuations, vector spaces, Hamel bases, quotient spaces, convexity, linear operators, linear functionals, hyperplanes, and the algebraic Hahn-Banach theorem. It then develops normed and Banach spaces, completions, finite-dimensional normed spaces, the Baire theorem, Schauder bases, separability, bounded linear operators, the Hahn-Banach theorem in normed spaces, duality, reflexivity, adjoint operators, and the fundamental theorems of functional analysis. Weak and weak-star convergence are treated from a sequential point of view. The final chapters present Hilbert space theory, compact operators, the Arzela-Ascoli theorem, integral operators, spectral theory, and the spectral theorem for compact self-adjoint operators. An appendix develops bounded variation and the Riemann-Stieltjes integral as tools for the representation of the dual of C([a, b]).

Key Features Include:

- A self-contained first course requiring no prior measure theory

- A gradual transition from algebraic foundations to operator theory

- Careful treatment of Hahn-Banach, duality, reflexivity, and the fundamental theorems

- A sequential approach to weak and weak-star convergence

- Exercises supporting both classroom use and independent study

The text is suitable for a first course and as a reference for readers seeking a precise foundation in modern analysis.

Zielgruppe

Undergraduate Advanced

Autoren/Hrsg.

Aduén, Hugo

Herrón, Sigifredo

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis

Inhaltsverzeichnis

1. Vector Spaces. 2. Normed Spaces. 3. Bounded Linear Operators. 4. Fundamental Theorems. 5. Weak and Weak* Convergence. 6. Hilbert Spaces. 7. Compact Operators and Spectral Theory.

Über Autor(innen)

Hugo Aduén is a full professor in the Departamento de Matemáticas y Estadística at the Universidad de Córdoba, Colombia. He received his Ph.D. in Mathematics from the Universidad Nacional de Colombia, Medellín, in 2005, under the supervision of Alfonso Castro and Jorge Cossio. His teaching has included functional analysis, measure theory, topology, and mathematical analysis. His research is centered on nonlinear analysis and, more specifically, on semilinear elliptic partial differential equations. He has obtained results concerning uniqueness and the existence of infinitely many nonradial solutions for elliptic problems.

Sigifredo Herrón is a full professor in the Departamento de Matemáticas at the Universidad Nacional de Colombia, Medellín. He received his Ph.D. in Mathematics from the same institution in 2006 under the supervision of Jorge Cossio. He teaches a variety of subjects, ranging from foundational calculus and linear algebra for engineering students to advanced courses in functional analysis, real analysis, and partial differential equations. He has successfully supervised many master's and doctoral theses. His research primarily focuses on nonlinear analysis, with a specific emphasis on semilinear and quasilinear elliptic partial differential equations, where he has obtained results regarding the existence, nonexistence, and properties of solutions.