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This text provides a broad foundation for the study of advanced mathematics and applications of linear algebra to other areas of mathematics. Featured application areas include linear ordinal differential equations, polynomial approximation, and finite element methods. It offers numerous exercises covering a range of difficulties.

Tikhonov regularization is the most popular general-purpose method for regularization, a mathematical technique to suppress the effect of noise in data, and uses much of the machinery of Hilbert space theory. This book develops the theory of Tikhonov regularization for a certain class of linear inverse problems which are defined on Hilbert spaces. To explain why and how Tikhonov regularization works, the singular value expansion for compact operators is introduced. Tikhonov regularization with seminorms is also analyzed and for this purpose, densely defined unbounded operators are addressed...