First, my definition of weak convergence in $X$ is that $x_n \rightharpoonup x$ if $\phi(x_n) \to \phi(x)$ for all $\phi \in X^*$.
I recently read the statement that $e_n \rightharpoonup 0$ in $\ell^p$, $p>1$, where $e_n$ is the canonical basis vector.
In $\ell^2$, this is clear to me, since the weak convergence $x_n \rightharpoonup x$ in $\ell^2$ (which is Hilbert) is equivalent to $\langle x_n, y \rangle \to \langle x, y \rangle$ for all $y \in \ell^2$.
But when $p \neq 2$, I struggle to prove this since I don't know the form of a general functional $\phi \in X^*$. Can you help me understand this?