This textbook, based on three series of lectures held by the author at the University of Strasbourg, presents functional analysis
in a non-traditional way by generalizing elementary theorems of plane geometry to spaces of arbitrary dimension. This approach
leads naturally to the basic notions and theorems. Most results are illustrated by the small lp spaces. The Lebesgue integral,
meanwhile, is treated via the direct approach of Frigyes Riesz, whose constructive definition of measurable functions leads
to optimal, clear-cut versions of the classical theorems of Fubini-Tonelli and Radon-Nikodym. Lectures on Functional Analysis
and the Lebesgue Integral presents the most important topics for students, with short, elegant proofs. The exposition style
follows the Hungarian mathematical tradition of Paul Erdos and others. The order of the first two parts, functional analysis
and the Lebesgue integral, may be reversed. In the third and final part they are combined to study various spaces of continuous
and integrable functions. Several beautiful, but almost forgotten, classical theorems are also included.
and graduate students in pure and applied mathematics, physics and engineering will find this textbook useful. Only basic
topological notions and results are used and various simple but pertinent examples and exercises illustrate the usefulness
and optimality of most theorems. Many of these examples are new or difficult to localize in the literature, and the original
sources of most notions and results are indicated to help the reader understand the genesis and development of the field.