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Finite-Dimensional Vector Spaces (Undergraduate Texts in by Paul R. Halmos

By Paul R. Halmos

From the experiences: "The concept is systematically constructed via the axiomatic procedure that has, considering that von Neumann, ruled the overall method of linear practical research and that achieves right here a excessive measure of lucidity and clarity....The ebook includes approximately 350 good put and instructive difficulties, which conceal a substantial a part of the topic. All in all this can be an exceptional paintings, of both excessive price for either pupil and teacher." --ZENTRALBLATT FÜR MATHEMATIK

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Prove that a subset of '0 is an equivalence class modulo ml: if and only if it is a coset of ml:. 4. (a) Suppose that ml: is a subspace of a vector space '0. , an element of '0') j the linear functional z is defined by z(x) = y(x + ml:). Prove that the correspondence Y -+ Z is an isomorphism I between ('O/ml:)' and ml:0 • (b) Suppose that ml: is a subspace of a vector space '0. , an element z of ml:') j the linear functional z is defined by z(x) = y(x). Prove that z is unambiguously determined by the coset JC (that is, it does not depend on the particular choice of y), and that the correspondence :Ie -+ z is an isomorphism between 'O'/ml:0 and ml:'.

It is, however, at this point that the greatest advantage of the notation [x, y] appears; by means of it, it is easy to discuss 'O and its relation to 'O". If we consider the symbol [x, y] for some fixed y = Yo, we obtain nothing new: [x, Yo] is merely another way of writing the value Yo(x) of the function Yo at the vector x. If, however, we consider the symbol [x, y] for some fixed x = Xo, then we observe that the function of the vectors in 'Of, whose value at y is [xo, y], is a scalar-valued function that happens to be linear SEC.

Warning: the assertion that a vector space is reflexive if and only if it is finitedimensional would shock most of the experts in the subject. The reason is that the customary and fruitful generalization of the concept of reflexivity to infinitedimensional spaces is not the simple-minded one given in (h). § 18. Direct SUll1S We shall study several important general methods of making new vector spaces out of old ones; in this section we begin by studying the easiest one. If 'U and '0 are vector spaces (over the same field), their direct sum is the vector space VI' (denoted by 'U E9 '0) whose elements are all the ordered pairs (x, y) with x in 'U and y in '0, with the linear operations defined by DEFINITION.

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