Prove Minkowski Inequality using Cauchy-Schwartz Inequality

[Rederick]

I expanded (x+y),(x+y) and got x^2+y^2 > 2xy then replaced 2xy with 2|x,y| but now I'm stuck.

I need to get it to ||x+y|| <= ||x|| + ||y||. Am I close?

 

[Rederick]

Here's what I did so far...

Let x=(x1,x2..xn) and y=(y1,y2..yn) in R. Assume x,y not = 0. Then (x+y)dot(x+y) = sum(x^2+2xy+y^2) >= 0. Then I rewrote it as sum(x^2 +y^2) >= sum(2xy). Using Cauchy Schwartz Inequality, sum(2xy) = 2|x dot y| <=2( ||x|| ||y||). So now I have this:

sum(x^2) +sum(y^2) <= 2( ||x|| ||y||).

I'm not even sure I'm doing the right thing. Can anyone help?

 

[Fredrik]

Start with $\|x+y\|^2$. This is less than what?

I'm assuming that what you're trying to prove is that

$$|\langle x,y\rangle|\leq \|x\|\|y\|\Rightarrow \|x+y\|\leq\|x\|+\|y\|$$

I'm not familiar with the term "Minkowski inequality". I would call the inequality on the right the "triangle inequality". (Edit: Aha, it's the triangle inequality for a specific Hilbert space).

By the way, if you click the quote button next to this post, you can see how I did the LaTeX. Keep in mind that there's a bug that causes the wrong images to appear in previews most of the time, so you will need to refresh and resend after each preview.

 

[Rederick]

Thank you Fredrik. I got it.