Schwarz inequality is Cauchy–Schwarz inequality?

rfrederic

I found many information showed Schwarz inequality and Cauchy–Schwarz inequality are same on books and internet, but my teacher's material shows that:
Schwarz inequality:
[itex]\left\|[x,y]\right\|\leq\left\|x\right\|+\left\|y\right\|[/itex]

Cauchy–Schwarz inequality:
[itex]\left\|[x,y]\right\|\leq\left\|x\right\|\left\|y\right\|[/itex]


They seem to be different on material, and I had sent email to teacher but having no reply.
Therefore my question is "Are Schwarz inequality and Cauchy–Schwarz inequality same?"

mfb

Those do not look like inequalities to me. And the first one looks wrong, independent of the inequality sign.
Maybe you mean ##||x+y|| \leq ||x||+||y||##, but that is the triangle inequality. It follows from the Cauchy–Schwarz inequality if the norm is induced by a scalar product.

rfrederic

Sorry about used wrong symbol, and I have modified.
I know triangle inequality.

But the question still is "are Schwarz inequality and Cauchy–Schwarz inequality same?"


Thanks for your reply.

micromass

Your Schwarz inequality simply seems false. In [itex]\mathbb{R}[/itex], we have [itex][x,y]=xy[/itex]. But it is certainly not the case that

[tex]|2\cdot 3|\leq |2|+|3|[/tex]

Vargo

It looks like a typo to me. The books and the internet are right I think.

rfrederic

Well, do you mean that:
Schwarz inequality
= Cauchy–Schwarz inequality
= [itex]\left\|[x,y]\right\|\leq\left\|x\right\|\left\|y\right\|[/itex]?

I think so either, thus I want to figure it out.


Both of your answers are helpful, thanks.