Proving that $\left\lVert x \right\rVert_a := \sum_{k=1}^{\infty}\ |x_k| $ defines a norm on a set

Question

I have a set $a:=$ {$x \in \mathbb{C}: \sum_{k=1}^{\infty}\ |x_k| < \infty $}.

How can I prove, that $\left\lVert x \right\rVert_a := \sum_{k=1}^{\infty}\ |x_k| $ defines a norm on a?

I know that norms can also be defined on the vector space $\mathbb{C}^{\mathbb{N}} := {x = (x_k)_{k\in \mathbb{N}}: x_k \in {\mathbb{C}}}$ for all $k \in \mathbb{N}$.

So if $x,y \in \mathbb{C}$ and ${\lambda \in \mathbb{C}}$, then there is the addition and scalar multiplication in the vector space, which are defined as

$x + y := (x_k + y_k)_{k\in \mathbb{N}}$ and $\lambda x := (\lambda x_k)_{k\in \mathbb(N)}.$

I didn't have much to do with infinite dimensional vector spaces, I've hear of $\mathbb{C}^1(\mathbb{R})$ or $\mathbb{C}^0(\mathbb{R})$, but that's where it ends..

Answer 1

You see it'a norm just by checking the definition, if you have any issue you can check in the litterature: just search for $l^1$ space (note: $l^1$, not $L^1$).