By C. H. Edwards Jr.

Modern conceptual remedy of multivariable calculus, emphasizing the interaction of geometry and research through linear algebra and the approximation of nonlinear mappings via linear ones. whilst, plentiful cognizance is paid to the classical purposes and computational equipment. 1000's of examples, difficulties and figures. 1973 edition.

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If S is an infinite subset of A, then S has a limit point be B, because B is compact. But be A also, because b is a limit point of A, and A is closed. Thus every infinite subset of A has a limit point in A, so A is compact. | In the next theorem and its proof we use the following notation. Given x = (xl9 . . , xj e 0Γ and y = (yl9 . . , yn) e mn9 write (x, y) = (xl9 . . , xm,yl9 . . , yn) e @m+n. If A c ^ m and 5 c @n9 then the Cartesian product A x B = {(a, b)e@m+n:*eA is a subset of @ and b e B} m+n .

D) (rA)B = r(AB) = A(rB). PROOF We prove (a) and (b), leaving (c) and (d) as exercises for the reader. Let the matrices A, B, C be of dimensions k x /, / x m, and m x n respectively. Then let f:&l^&k,g: Mm -► M\ /? : &n -► Mm be the linear maps such that Mf = A,Mg = B, Mh = C. Then (/o(0oA))(x) =f(goh(x)) = f(9(h)x)))=(fog)(h(x)) = ((fog)oh)(x) n for all x e M , so f° (g ° h) = (f ° g) ° h. 2 therefore implies that A(BC) = MfMgûh = Mfo(goh) = (MfMg)Mh = (AB)C, thereby verifying associativity. To prove (b), let A be an / x m matrix, and B, C m x n matrices.

Show that A is compact if and only if every sequence of points of A has a subsequence that converges to a point of A. If |b„ | > n for each n, show that the sequence {b„}T has no limit. Prove that the union or intersection of a finite number of compact sets is compact. Let {An}™ be a decreasing sequence of compact sets (that is, An + 1 <= A„ for all n). Prove that the intersection Π^°=ι ^« ^s compact and nonempty. Give an example of a decreasing sequence of closed sets whose intersection is empty.