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Question 6. For vectors in x e Rd, define d ||2||1 = 12; j=1 d 1/2 j=1 ||1||2 = (22;?)" || 2 || 1 = maxlu; | For any dimension d, and vector ce Rd, show that (1) ||2||2 < ||2||2 < ||2||1 (WLOG, you can assume || 2 || 1 = 1. Why?) (2) ||2||1 5 V2||2||2. And show that the inequality is sharp in general. (Cauchy-Schwartz) (3) || 2 ||1 <d||2||20- And, show that the inequality is sharp in general.
Question 7. Using the same notation as above, answer these questions with an example or a proof that there is no such vector x e Rd. The answers can depend upon dimension d. (1) || 2 ||1 = 1 while ||0|| 20 = 2 (2) || 2 || 1 = 1 while || 2 || 2 = V2 (3) || 20 || 1 = 100 while || 0 ||. = 1 (4) || 2 || 1 = 100 while || 2 ||2 = 1