Difference between revisions of "Talk:Cauchy-Schwarz Inequality"
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Any two vectors define a plane; in that plane, we can measure the angle between them, and CS is then equivalent to the fact that the cosine of this angle is less than 1 in absolute value. This is just as much a proof of CS (taking for granted some simple facts about the geometry of <math>{\bf R}^n</math>) as it is the reverse -- if you think these facts about <math>{\bf R}^n</math> are "less basic" than CS, feel free to add a proof of CS using "more basic" things. --[[User:JBL|JBL]] 15:58, 9 April 2008 (UTC) | Any two vectors define a plane; in that plane, we can measure the angle between them, and CS is then equivalent to the fact that the cosine of this angle is less than 1 in absolute value. This is just as much a proof of CS (taking for granted some simple facts about the geometry of <math>{\bf R}^n</math>) as it is the reverse -- if you think these facts about <math>{\bf R}^n</math> are "less basic" than CS, feel free to add a proof of CS using "more basic" things. --[[User:JBL|JBL]] 15:58, 9 April 2008 (UTC) | ||
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| + | Hm. I agree with you that the cosine way is indeed a valid proof. But both of the proofs of the general form on this page are extremely close to the axioms of fields and inner product spaces, using almost no intermediate results, whereas the axiomatic construction of <math>\mathbf{R}^n</math> and derivation of basic trigonometry is pretty nontrivial by comparison. So while I agree with you that it is perfectly valid to go this way, I also think it is valid (and perhaps easier) to construct <math>\mathbf{R}^n</math> and define the cosine function analytically, and then to prove the restrictions on its range using one of the general proofs of Cauchy. —[[User:Boy Soprano II|Boy Soprano II]] 18:40, 9 April 2008 (UTC) | ||
Revision as of 14:40, 9 April 2008
USAMO 1995 number 5 is a great problem solving example.--MCrawford 15:22, 18 June 2006 (EDT)
"not really a proof"
Any two vectors define a plane; in that plane, we can measure the angle between them, and CS is then equivalent to the fact that the cosine of this angle is less than 1 in absolute value. This is just as much a proof of CS (taking for granted some simple facts about the geometry of 
) as it is the reverse -- if you think these facts about are "less basic" than CS, feel free to add a proof of CS using "more basic" things. --JBL 15:58, 9 April 2008 (UTC)
Hm. I agree with you that the cosine way is indeed a valid proof. But both of the proofs of the general form on this page are extremely close to the axioms of fields and inner product spaces, using almost no intermediate results, whereas the axiomatic construction of 
and derivation of basic trigonometry is pretty nontrivial by comparison. So while I agree with you that it is perfectly valid to go this way, I also think it is valid (and perhaps easier) to construct and define the cosine function analytically, and then to prove the restrictions on its range using one of the general proofs of Cauchy. —Boy Soprano II 18:40, 9 April 2008 (UTC)
