Looking for a general proof of the Cauchy-Schwarz Inequality

Asked Jun 15 '21 at 06:28

Active Jun 15 '21 at 06:28

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I was reading the book Mathematics for Machine Learning by Marc Peter Deisenroth, Cheng Soon Ong and A.Aldo Faisal and came across this remark:

Remark(Cauchy-Schwarz Inequality). For an inner product vector space $(V,\langle\cdot,\cdot\rangle)$, the induced norm $||\cdot||$ satisfies the Cauchy-Schwarz inequality
$|\langle x,y\rangle|\le ||x||\text{ }||y||$

Now I know how to prove this inequality for the dot product and the induced Euclidean norm.

But I want to know if there is some general proof of this inequality that can show the truth of this statement for every possible inner product and induced norm.

So, my question is whether such a proof exists, and if yes how it works(some clues, not the proof itself), or whether the inequality has to be proven separately for any considered inner product.

Thanks for your time!

asked Jun 15 '21 at 06:28

Gaurav Chandan

1

There are a couple of proofs in the wikipedia page for the Cauchy-Schwarz inequality. – Amit Rajaraman Jun 15 '21 at 06:32
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Once you have proved it for a general inner product space (over $\mathbb R$ or $\mathbb C$), then you don't have to re-prove it for specific inner products. That's the power of the abstract definition of inner product! – Jun 15 '21 at 06:38
Thank you people! – Gaurav Chandan Jun 15 '21 at 12:45

Looking for a general proof of the Cauchy-Schwarz Inequality

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