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I've started to study metric spaces recently and got intrigued with a demonstration which led me to a bunch of ideas and questions. First of all, I've proved the Cauchy inequality and then Cauchy-Schwarz. Both of then - at least in every reference I found - were lying on the very-well known property of quadratic equations, the discriminant. You end up solving the problem of demonstrating them making use of that.

But now I faced an exercise which suggested a norm defined by:

$|\space|_p:E\times E \rightarrow R$ $$ |x|_p=\big(\sum_{i=0}^{m}|x_i|^p\big)^{1/p} $$ Well, the exercise just ask me for some application of this norm, but I got interested in trying to demonstrate it`s actually a norm and got in a bunch of trouble when trying to prove the triangular inequality. What I tried to do was to proceed just the same way I did for the inner product (Cauchy-Schwarz): $$(|x+y|_p)^p=\sum_{i=0}^{m}|x_i+y_i|^p$$ $$(|x|_p+|y|_p)^p=\sum_{i=0}^{m}(|x_i|+|y_i|)^p$$ Ok, first question: Is there a way of expanding the first summation inner term into a sort of binomial thing? I did this with the second but couldn't figure out if it was possible to do in the first one.

Next question:Well, suppose I could proceed in that way. How would proceed the same way I did for Cauchy-Schwarz if it will not likely end up in a quadratic equation which has a discriminant as criteria for building up the inequality?

Final question: Is there any generalization of Cauchy-Schwarz`s which would fit for that p power thing?

Hope I was clear enough but I put myself at your disposal for any elucidation needed. Thanks in advance, Marcelo.

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    $\begingroup$ Your second equation is false. Usually, one first proves the Hölder inequality (that is the generalization of Cauchy Schwarz you are looking for) and uses it in the proof. The whole thing is not straight forward. Therefore, I suggest you look into a book where the proof is demonstrated. $\endgroup$ – Friedrich Philipp Mar 8 '16 at 1:14
  • $\begingroup$ Can you indicate some reference? $\endgroup$ – Marcelo Broinizi Mar 8 '16 at 1:51
  • $\begingroup$ The inequality you want to prove is called Minkowski's inequality. Wikipedia should be sufficient as a reference. $\endgroup$ – Friedrich Philipp Mar 8 '16 at 2:31
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may be interested for you: Danko R. Jocić 1, Stefan Milošević, Refinements of operator Cauchy–Schwarz and Minkowski inequalities for p-modified norms and related norm inequalities,Linear Algebra and its Applications 488 (2016) 284–301

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I will just answer your final question for now, because it is a key intermediate step in proving the triangle inequality in the $\ell^p$-norm you describe.

The following generalisation of Cauchy's inequality is true:

For $p,q\geq 1$ such that $1/p+1/q=1$, we have

$$(\sum_{k=1}^n |x_k|^p)^{1/p}(\sum_{k=1}^n |y_k|^q)^{1/q}\geq \sum_{k=1}^n |x_ky_k|.$$

This is called Hölder's inequality, and is the generalisation of Cauchy--Schwarz (We recover C-S if we set $p=q=2$) that you want to prove that $\ell^p$ norms are norms.

In order to prove Hölder's inequality I provide the following hint:

Note that scaling the vectors $x,y$ by a positive constant multiplies both sides of the equation by the same factor. (And so preserves the inequality.)

Can you choose a scaling that simplifies the ugliest part of the inequality you are trying to prove?

After doing this, try to prove the triangle inequality using Hölder's.

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