Why is every $l^p(\mathbb{K})$ a dual of some normed space?
The $l^p(\mathbb{K})$ space is the space:
$$\{ w \in F(\mathbb{N}, \mathbb{K}) : \sum_{n=0}^{\infty} |w(n)|^p < \infty\}$$
where $F(\mathbb{N}, \mathbb{K})$ is the space of functions $f: \mathbb{N} \rightarrow \mathbb{K}$.
Is it simply because the duals of normed spaces are continuous functions (or bounded linear) and functions of $l^p$ are a type of continuous functions? But how do I know that $w$s are continuous?