State and prove conditions for $\|x\|_a=\sum_{j=1}^n a_j\lvert x_j\rvert$ to be a norm on $\mathbb R^n$

Question

Let $a_j \in \mathbb R$ for $1\leq j \leq n$. State and prove necessary and sufficient conditions for $\|\cdot\|_a$ to be a norm on $\mathbb R^n$.

I know the properties needed for a function to be a norm so I don't know if I'm meant to state them to prove that the given function is a norm or if I need to list some extra requirements for my specific function to be a norm. If so then how would I prove them to be necessary and sufficient?

Answer 1

Hint

Prove that $a_j>0,\; \forall 1\le j\le n$ is the desired necessary and sufficient condition.

Answer 2

The question comes down to the following: if we know that $$ \sum_j a_j|x_j| = 0 \implies x_j = 0 \text{ for all }j $$ and if we know that $$ \sum_j a_j|x_j + y_j| \leq \sum_j a_j|x_j| + \sum_j a_j|y_j| $$ for every set of vectors $x$ and $y$, then what can we deduce about the values of $a_1,\dots,a_n \in \Bbb R$?