Cauchy-Schwartz inequality

Dear Sir,

For question 2 in this week’s sheet, I have no idea what exactly we are trying to prove! Can you help??


Recall that the Cauchy-Schwartz inequality says |< v,w>| is less than or equal to ||v||||w|| for any two vectors v and w in an inner product space. The problem is asking you to prove that in fact

|< v, w>|< ||v||||w||

that is, that there is a *strict* inequality, except in the case where one vector is a multiple of the other.

If you examine the proof I gave in the lecture, you’ll see that this statement is actually hidden in there.



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