How to show that a Banach space is a subspace of another Banach space?

Question

I'm having some trouble showing that the following Banach space is a subspace of $\ell^1(\mathbb{N} )$

$\ell^1_w(\mathbb{N} ) = \{ \{x_k\}^\infty_{k=1}| x_k \in \mathbb{C}, \sum_{k=1}^{\infty} |x_k|\cdot2^k < \infty \}$

I hope someone can guide me in the right direction.

Thanks in advance.

Answer 1

Let $x \in \ell^1_w$. Then we have, as $1 \le 2^k$ for every $k$ that $$ \sum_k |x_k| \le \sum_k 2^k |x_k| < \infty $$ that is $x \in \ell^1$. Therefore $\ell^1_w \subseteq \ell^1$, it remains to show that it is a subspace, that is $x+\lambda y \in \ell^1_w$ for every $x,y \in \ell^1_w$, $\lambda \in \mathbf K$.