Proof of a p-norm and Cauchy/Cauchy-Schwarz inequality generalization 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.
 A: 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.
A: 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
