Because of the crucial role played by functional analysis in the applied sci-
ences as well as in mathematics, I have attempted to make this book ac-
cessible to as wide a spectrum of beginning students as possible. Much of
the book can be understood by a student having taken a course in advanced
calculus. However, in several chapters an elementary knowledge of functions
of a complex variable is required. These include Chapters 6, 9, and 11. Only
rudimentary topological or algebraic concepts are used. They are introduced
and proved as needed. No measure theory is employed or mentioned.
The book is intended for a one-year course for beginning graduate or
senior undergraduate students. However, it can be used at any level where
the students have the prerequisites mentioned above.
I have restricted my attention to normed vector spaces and their impor-
tant examples, Banach and Hilbert spaces. These are venerable institutions
upon which every scientist can rely throughout his or her career. They are
presently the more important spaces met in daily life. Another considera-
tion is the fact that an abundance of types of spaces can be an extremely
confusing situation to a beginner.
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