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Review title: Excellently organized book.
Product review: This book presents various mathematical principles in an organization I have not seen before. It starts with the idea of a transformation, then goes on to relate eigenvalues and eigenvectors to general spectral theory, explain how the need for closed function spaces naturally leads to Lebesgue integration (I know about Lebesgue integration before but I didn't know why it was needed), and show how the definition of certain inverse operators leads to distribution theory. This is a very natural way of organizing these principles. While other books, such as Strang's Intro to Applied Mathematics and Rudin's Real & Complex Analysis, provide you with one mathematical "toy" after another (Fourier series, Lebesgue integration, etc.), Keener's book tells you why you need the toy before giving it to you.
