X Preface Second, the writing itself tends to be concise and to the point, to the extent that some of the proofs might be better described as detailed lists of hints. This is intentional—we believe that students learn more by having to fill in some details themselves. Besides it s style , this book differ s fro m man y othe r text s o n the subject in that we try to present the main ideas, whenever possible, in the context of vector spaces over a general field, F, rather than assum- ing the underlying field to be R or C. Inner-product spaces, along with the naturally associated classes of self-adjoint, normal , and unitary (or orthogonal) operators , ar e introduced late r tha n i n many books, an d the spectral theorems for these operators, besides being fundamentall y important on their own, also serve here to pave the way for the notions of reducing and semisimplicity and, eventually, to the general structure theorems—the Jordan form, when the underlying field is algebraically closed, and the corresponding form over general fields. The tex t consist s o f eigh t chapter s an d a n appendix . Thes e ar e divided into sections, and further into subsections. Definitions, propo- sitions, examples , etc. , ar e numbered accordin g t o the subsectio n i n which they appear, and no subsection has more than one object (defi - nition, theorem, etc.) o f each kind. Fo r example, Lemma 1.3. 5 i s the lemma appearin g i n subsectio n 1.3.5 , and Theorem 1.3. 5 i s the the- orem appearing i n the sam e subsection . Reference s t o the appendi x have the form A.x.y (for subsection y of section x, in the appendix). Exercises appear at the end of sections, and are numbered accord- ingly, e.g., exercise ex3.1.2 is the second exercise of section 3.1. Starred sections , subsections , an d exercises contai n materia l that can be skipped on first reading. Severa l of these sections, as well as parts of the additional topics (in Chapter 8) , require some familiarit y with basic analysis, e.g., concepts like convergence and continuity.
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