This book brings together all available results about the theory of algebraic multiplicities, from the most classic results, like the Jordan Theorem, to the most recent developments, like the uniqueness theorem and the construction of the multiplicity for non-analytic families. Part I (first three chapters) is a classic course on finite-dimensional spectral theory, Part II (the next eight chapters) presents the most general results available about the existence and uniqueness of algebraic multiplicities for real non-analytic operator matrices and families, and Part III (last chapter) transfers these results from linear to nonlinear analysis. The text is as self-contained as possible and suitable for students at the advanced undergraduate or beginning graduate level.

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