When does the Cauchy-Schwartz inequality hold?

[Benny]

Hi, I need to use the Cauchy-Schwartz inequality to prove the following inequality.

<br /> \left( {a_1 + ... + a_n } \right)^2 \le n\left( {a_1 ^2 + ... + a_n ^2 } \right),\forall a_i \in R<br />

When does equality hold?

The Cauchy-Schwartz inequality is \left| {\left\langle {\mathop u\limits^ \to ,\mathop v\limits^ \to } \right\rangle } \right| \le \left\| {\mathop u\limits^ \to } \right\|\left\| {\mathop v\limits^ \to } \right\|.

The Cauchy-Schwartz inequality holds for all inner products. Since the dot product is the only 'standard' inner product then the dot product is probably going to be needed here which hopefully means that the calculations won't be too involved.

At the moment I'm lost for ideas. Seeing the n on the RHS suggests that the two vectors have components which are multiples of n and reciprocals of n. Also, seeing that only a_i appears on both sides of the equation. I think I can take u = v in the Cauchy Schwartz inequality. So I'm dealing with a single vector. I can't think of a way to start this. Does anyone have any suggestions? Any help would be great thanks.

 

[AKG]

Yes, think about the standard inner product, and find a specific vector u such that the thing you're trying to prove is just a statement of the Cauchy-Schwartz inequality for that specific vector u. For an analogy, what you're trying to prove is analogous to proving |5 + y| < |5| + |y| given the triangle inequality |x + y| < |x| + |y|. Once you see how your inequality is related to the Cauchy-Schwartz inequality, you will be able to determine when equality holds in your inequality from when it holds in the C-S inequality.

Hint, square root both sides of your inequality (or square both sides of the C-S inequality) and recall the definitions of the standard norm and the standard inner product.

 

[Benny]

Thanks, I'll have a think about it.