M 311 Spring 2004 HOMEWORK SUMMARY REPORT Chapter number: 6 Section number: 1 Exercise number: 3 Number of papers received: 4 Reviewing committee: Theta List all participating committee members: Xujia Zhang, Jason Madsen Author(s) of paper(s) chosen for publication: Kathryn Turner Comments: We received four papers for this problem. Two of the solutions have some minor mathmatically mistakes (left out the square root signs in the second step for both of them). The other two are correct (one of them needs more mathmatically derviations and illustrations). We chose to publish Kathryn Turner's paper because hers is very well written and easy to follow. GRADER'S COMMENT: Several people seemed to think that bilinearity implies = + . Instead, using bilinearity twice, you have = + = + 2 + . INSTRUCTOR'S COMMENT: Most papers were organized by writing down the inequality to be proved and then manipulating it into something that is obviously true. That is probably the easiest way to FIND a proof; however, PRESENTING the proof that way can be confusing, for two reasons. One, it may not be clear to the reader (unless you write "?" above the inequality signs) which statements are being investigated rather than asserted. Second, both you and the reader may get mixed up about the direction of the logic. In the present case, the inequality to be proved can be shown to be equivalent to the Cauchy-Schwarz inequality without absolute value signs on the left side. This does not imply the inequality with the absolute value, but it is IMPLIED BY it, which is what is needed to prove the theorem!