This textbook contains the course of lectures about some concepts in functional analysis for Ph.D. students of engineering. Functional analysis is a study of abstract linear spaces resulting from a synthesis of geometry, linear algebra and mathematical analysis. Some of the calculations of this textbook were made using symbolic-numeric computer algebra system Mathematica
. The author found this system very useful for presenting these concepts because it does algebra and calculus computations quickly in an exact, symbolic manner.
The book also contains an easy introduction into the Lebesque measure and integration theory.
Vector spaces, subspaces, linear manifolds | Dimension, spanning sets and (algebraic) basis | Linear operator | Normed spaces | Convergence, complete spaces | Continuous and bounded linear operator | Dense sets, separable spaces | Inner product, Hilbert space | Sets of measure zero, measurable functions | The space L2 | Generalized derivatives, distributions, Sobolev spaces | Weak (or generalized) solutions | Orthogonal systems, Fourier series | The projection theorem, the best approximation