Etude Géométrique des Espaces Vectoriels: Une Introduction by Dr. Jacques Bair, Dr. René Fourneau (auth.)

By Dr. Jacques Bair, Dr. René Fourneau (auth.)

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Extra info for Etude Géométrique des Espaces Vectoriels: Une Introduction

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AUS I C bA C ~ (IV. 1). ,n-1, i ~o\I1,... ,r I ; 0 < x j < 1 (j:1, .... r) ; 2 Xk+ I + (Xk_1)2 ~ I , V k ~ ]Nol Soit x ~ ~ tel que x n = 0 (donc x I = x 2 = ... = X n _ I = 0). Pour tout n-1 Y @ b A, la droite (y:x) coupe IX : X n+1 2 + (Xn_ I)2 =I I en x et en 2Y n x' = x + 2 2 (y-x), ces d e u x points 4tant distincts, puisque Y n + Yn-1 Y n ~ 0, y ~tant un p o i n t de bn'1 A. Comme 2 IX : X n+1 + (Xn_ I)2 < 11 n-1 est c o n v e x e , i l c o n t i e n t le s e g m e n t [x:x'] donc [y:x[ 0 C b A D ]x:x'[ n n-1 n bn A, on o b t i e n t et x ~ b A.

41 et, s i x ~ ~1([a:b[), : + , , et x = (~o~)(x) = k;1(a) soit X ~ [ ~ 1 ( a ) : ~ ( b ) [ . La preuve e st semblable + (~-k) ;~(b) , 0

J de I. Si A o Sinon, [k:x[ O est une demi-droite poin- O ( iAo ( A. I). DSs lots, C n A. e t x ~ b( n A . ) . j~I J j6I J L ' a t t e n a n c e de i t i n t e r s e c t i o n de p a r t i e s a l g @ b r i q u e m e n t i r r a d i @ e s sur un nSme point ne oo~'ncide pas t o u j o u r s avec llin t e r s e c t i o n des a t t e n a n c e s de ces p a r t i e s . Pour preuve, dans ~2, si A l = i ( x l , x 2 ) aA I O a A 2 = vaut pas point 6m2:xl =01 et A2 = { ( x l , x 2 ) {(0,0)}; pour des (consid4rer tandis que ensembles a(A I Q A 2 ) E:~2:~2 : 01, = ~.

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