Principles of Mathematical Analysis
نویسندگان
چکیده
3. (a) If F is as in Theorem 10.7, put A = F′(0), F1(x) = A −1F(x). Then F1(0) = I. Show that F1(x) = Gn◦Gn−1◦. . .◦G1(x) in some neighbourhood of 0, for certain primitive mappings G1, . . . ,Gn. This gives another version of Theorem 10.7: F(x) = F(0)Gn◦Gn−1◦. . .◦G1(x). (b) Prove that the mapping (x, y) 7→ (y, x) of R onto R is not the composition of any two primitive mappings, in any neighbourhood of the origin. (This shows that the flips Bi cannot be omitted from the statement of Theorem 10.7.)
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