Volumes I through V of Theorems and Problems in Functional Analysis: The Answer Book present different techniques for solving the 828 Exercises found in the A.A. Kirillov and Gvichiani book, entitled Theorems and Problems in Functional Analysis. The original book by Kirillov and Gvichiani was in the Problem Book in Mathematics Series, published by Springer.

This is the most important and rich volume as it focuses on topological vector space and linear operators.  This volume covers: convexity, seminorms and Minkowski functions, duality, weak and strong topology, the Hahn Banach theorem and Banach spaces, the Helly theorem, compact sets theory, the Krein-Milman theorem, Fredholm operators, cohomology, Hilbert-Schmidt operators, distributions, functional spaces, the Stone-Weierstrass theorem, smooth functions, locally convex spaces, the dirac distribution, the (Schwarz) kernel theorem, distrbutional derivative, Hilbert Spaces, Rademacher functions, Walsh functions, Haar functions, between principle in Hilbert spaces, and much more.

Supplementing the book, Theorems and Problems in Functional Analysis, these volumes may be used by graduate students taking a course in functional analysis.  In order to understand this text, the reader must be familiar with mathematical analysis and real analysis.