By Edwards, Charles Henry

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**Sample text**

Show that (S, ) is a vector space with the operations defined in Example 6. Note that ({1, . . , n}, ) can be interpreted as n since the function may be regarded as the n-tuple (φ(1), φ(2), . . , φ(n)). 2 SUBSPACES OF n In this section we will define the dimension of a vector space, and then show that n has precisely n − 1 types of proper subspaces (that is, subspaces other than 0 and n itself)—namely, one of each dimension 1 through n − 1. In order to define dimension, we need the concept of linear independence.

With a = c = 1, b = 0 we obtain the usual inner product on 2. An inner product on the vector space V yields a notion of the length or “size” of a vector , called its norm x. In general, a norm on the vector space V is a real-valued function x → x on V satisfying the following conditions: for all and . Note that N2 implies that 0 = 0. The norm associated with the inner product , on V is defined by It is clear that SP1–SP3 and this definition imply conditions N1 and N2, but the triangle inequality is not so obvious; it will be verified below.

A proper emphasis on these problems, and on the illustrative examples and applications in the text, will give a course taught from this book the appropriate intuitive and conceptual flavor. I wish to thank the successive classes of students who have responded so enthusiastically to the class notes that have evolved into this book, and who have contributed to it more than they are aware. In addition, I appreciate the excellent typing of Janis Burke, Frances Chung, and Theodora Schultz. Advanced Calculus of Several Variables I Euclidean Space and Linear Mappings Introductory calculus deals mainly with real-valued functions of a single variable, that is, with functions from the real line to itself.