Question

Explain this Theorem as if you talking to a 5 year old

Theorem 22 For any pe2, p-* c,e,. defined by (6-57) where the numbers cr are c.-p.en The proof of this is not quite as trivial as it looks. If we start with p-(a1, a2, ), then (6-57) immediately gives a.-c" for all n, and all that remains is to show that (6-58) Cn e The only question lies in knowing what the right side means. As with series of numbers, we interpret the right side as limy-. Sv where Sy , cne," but “limit', must now be understood in the sense of the metric of 2, meaning that limpS0. However, since S is a finite sum, S- (c1. c2,.... cv. 0,0, 0,.), and Since this is the tail of the convergent series indeed, limy-.pS0, as required. a? we see that

Answer 1

As you mentioned that you want the proof of theorem like a 5 years old child so 9 am trying to give you a very simple explanation For any vector p elementof l^2 To prove that P = sigma^infinity_n = 1 c_n e_n where c_n = P. en Proof: let P = (a_1, a_2, --- a_n, ---) elementof l^2 and here e_i = (0, 0.... 1, 0, 0, .... 0,.....) Therefore e e_i = a_i implies a_n = P e_n for all n \r\n Proof: let P = (a_1, a_2, ...... a_n .....) elementof l^2 and here e_i = (0, 0, ......., 1, 0, 0, ...... 0, .....) Therefore p e_i = a_i implies ith place a_n = p. e_n for all n Now P = sigma^infinity_n = 1 a_n e_n = a_1e_1 + a_2e_2 + a_3e_3 + ....... = c_ne_1 + c_2e_2 + c_3e_3 + ..... = sigma^infinity_n = 1 c_ne_n Hence The result