I have to prove, that a sum of two norms, is a norm: $$\|\cdot\| := \|\cdot\|_a + \|\cdot\|_b.$$
Properties to satisfy:
$ (a)~ \|x\|=0 \iff x=0 $
$ (b)~ \|\alpha x\|=|\alpha| \|x\| $
$ (c)~ \|x+y\| \le \|x\|+\|y\| $
I'm not sure how to show all of these properties. I mean, it sounds logical and I know it's true, but I have a blank space in my mind.
Remember to prove this you need to assume that $\|\cdot\|_{a}$ and $\|\cdot\|_{b}$ are both norms, and thus satisfy the definition. Thus implies what you need to prove.
One puts in an abstract vector $x$, and show the equality of the statements.
For example: For the first part of the definition, if $\|x\|= \|x\|_{a} + \|x\|_{b} = 0$, what can you say about $\|x\|_{a}$ and $\|x\|_{b}$ (based on the fact that $\|\cdot\|_{a}$ and $\|\cdot\|_{b}$ both satisfy this part of the definition)?