By Howard Eves
From the book's preface:
Since writing the preface of the 1st version of this paintings, the gloomy plight there defined of starting collegiate geometry has brightened significantly. The pendulum turns out certainly to be swinging again and a goodly volume of fine textual fabric is showing.
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Extra info for A Survey of Geometry (Revised Edition)
By the properties of g(z), it follows that for any positive number E there exist numbers 7 > 0 and n 2 1 such that Ilevz'gn(z)- g(z)IIF(r)S E* Hence it wiU be sufficient to approximate functions of the form eazagn(z). If z = x + i y , 0 < x < 1 , then + iy) = c . i- bk,n(x) nnm m=1,2, .... - nnm The last expression is independent of m, since gn(x + iy) has the period 2nn in ydirection. The series ( 7 ) converges in A , + A , in the weak topology. Taking m + co, if follows by (8) and Cauchy's theorem that ak,n(x) = a k , n is independent of x.
Ii'(t)dt = ii(0) - lim c ( N ) 0 N-. m 0 exists in A , + A , . The last limit exists in the space A , . Since tW belongs to J ~ ; ~ ( A , ) , this limit must be zero. Hence by (5) and (6), ~ ~ ~ ( P 0 ~ 9 0 ~~L 0 A~l )P, 1 7llalls ~ 1 s cllallz-,'. Consequently, G ( P o 9 To, A,; PI, A,) = JYPO,To, A,; P l , % - 1 , A l ) . (7) Step 3. Let qo > 0, ql < 1, m = 1, j = 0, and a E X ( p o , q o ,A , ; p , , q1 - 1, A,). Then we have Gc a = sw(T) dt 7 9 lIt'ovllL;o(A,) + lIt'l-lVllL~l(A1) 0 Let 00 [ u(t) = t w(t) dt 7= 12);,( 1 0 $.
Formulas (5) and (6) yield that u@-l)(t)is absolutely continuous on (0, 00) and a = u@-l)(l). Formula (3) shows that t"+-L(m) E L,((o, I ) , A,) (7) is valid. tAl) < l I ~ ( ~ - l )IIA,( +lA, ) + ~ l l t " u (I1~L;,(A,) ) (8) holds. +A, + cllull Vm* For tfhe proof of (4), we still must estimate the first term in (9). 8. The Trace Method Let 27 94t) E ~ @ , 3 ) ) \, d t ) dt = 1 . 0 By multiplication of (10) with ~ (-ls) and integration over s, there follows that 1 1 * 1 f 1 -f 7 Now, partial integration shows that 1 ~ ( ' ' ) (=l )J Y)(s)U ( S ) ds 3 1 + J x(z) u ( " ~ ) ( Td) t .
A Survey of Geometry (Revised Edition) by Howard Eves