E 00 (iii) = E 00 = 00 = E Proof. We show the following implications: (i) ==> (i) . (i) is, (x - y,ej) ==> -y (ii). Assume that = 0 for x is not one-to-one. Then (ii) ==> (iii) (x,ej) (y,ej) = ==> (v) ==> ==> x I- y, {e1, e2,... }. holds for I- 0 and this contradicts the completeness of S = (iv) that 1.
Invitation to Linear Operators: From Matrices to Bounded by Takayuki Furuta
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