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# Download e-book for iPad: Algebraic K-Theory and Algebraic Number Theory by Stein M.R., Dennis R.K. (eds.) By Stein M.R., Dennis R.K. (eds.)

ISBN-10: 0821850903

ISBN-13: 9780821850909

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Sample text

Let A be a nonempty subset of X. The convex hull of A, denoted by coCA), is the intersection of all convex sets that contain A. Verify the following statements. (1) coCA) is a convex set; hence coCA) is the smallest convex set that contains A. (2) A is convex if and only if A=co(A). Ai = 1, n EN}. 11. Let A be a nonempty subset of X. The conical hull of A, denoted by conCA), is the intersection of all convex cones that contain A. Verify the following statements. (1) con (A) is a convex cone; hence conCA) is the smallest convex cone that contains A.

Thus assume a =1= 0. Using (iii) we see that Yo E PaK(ax) if and only if Yo E aK and Ilax - yoll = d(ax, aK) if and only if ±Yo E K and lallix - ±yoll = lald(x, K). , Yo E aPK(x). (v) This is an immediate consequence of (ii), and (vi) follows from (iv). (2) (i) and (ii) follow from (1) (i) and (ii), and the fact that if M is a subspace and y E M, then M -+- y = M. (iii) and (iv) follow from (1) (iii) and (iv), and the fact that if M is a subspace and a =1= 0, then aM = M. The proof of (v) follows from the relations d(x-+-y,M) = mEM inf Ilx-+-y-mll= inf Ilx-+-y-(m-+-m')11 m,m'EM :s: m,m'ElvI inf [llx - mil -+- Ily - mill] = mEM inf Ilx - mil -+- inf Ily - mill = d(x, M) -+- d(y, M).

So {ad must be a bounded sequence in R Thus there exists a subsequence {ak j } and ao E lR such that akj --+ ao. 15), and the above argument shows that its limit is in M. This proves (1) and (2) when n = 1. Next assume that (1) and (2) are true when the dimension of Mis n and suppose M = span{xI,x2, ... ,xn,xn+d is (n+l)-dimensional. , ... ,Xn }, so Mn is n-dimensional. Since Mn is closed by the induction hypothesis and Xn+1 rf: M n , it follows that d(xn+1' Mn) > O. Let {Yk} be a bounded sequence in M.