A set $X=\lbrace x\in l^2:\Sigma_{k\geq1}k^2|x_k|^2<\infty\rbrace$. We need to prove that $\|x\|_X=\Sigma_{k\geq1}k^2|x_k|^2$ is a norm on set X. I tried to prove this through the four properties of the norm. But I have some difficulties of prove N3($\|\alpha x\|=|\alpha| \|x\|$) and N4($\|x+y\|\leq\|x\|+\|y\|$).
In N3, I prove $\|\alpha x\|=|\alpha|^2\|x\|$, which is wrong. But I don't know how to prove it.
In N4, I tried to use Cauchy-Schwarz inequality and Minkowski inequality. But I'm stuck.
Could anyone help me out here? Thanks in advance.