By Frank Ayres Jr.

ISBN-10: 9684229178

ISBN-13: 9789684229174

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Thus the range of p generates g x ff algebraically, so ijJ is unique if it exists. 07 (Prob. E). Since q> is bilinear, this extension annihilates the linear submanifold fJIl, and consequently can be factored through g x ff (Example L). 0 + Definition. Suppose ~: gj ~ ffj is a linear transformation, i = 1,2. 2. We shall call T the algebraic tensor product o{ TI and T2 and write T = TI X T2 . 1 has an important and useful counterpart that provides a categorical characterization of the algebraic tensor product.

1. A collection ~ of subsets of a set X forms a base for a topology on X if and only if every point of X lies in some element of ~ (briefly: if ~ covers X) and the intersection of every pair of sets belonging to ~ is a union of sets belonging to ~. ) Example A. The collection of all open intervals (a, b), where a, bE IR and a < b, is a base for the usual topology on IR. , the collection of all products of open intervals (ab bd x ... x (an, bn) is a base for the usual topology on IRn. A base for the usual topology on the complex plane C is given by the set of all open discs Dr(rt) = fA EC: Irt - AI < r}, where rt is a point of C and r is a positive number.

B. (i) Let X = {Xl' •.. , xm} and Y = {YI' ... , Yn} be linearly independent sets of vectors in a (real or complex) vector space g, and suppose X is contained in the submanifold j{ of g that is generated by Y. Show that m :-:; n and that it is possible to select a set Z of exactly n-m vectors from Y so that X u Z is also a basis for J/{. ) Conclude that if a vector space g possesses a finite basis, then any two bases for g contain the same number of vectors, and that, if dim g = n, then g is itself the only n-dimensional submanifold of g.

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