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 p 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-Nikodým. 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 Erdös 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. Both undergraduate 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.
