A First Course in Functional Analysis - Hugo Aduén

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]).

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