Convergence of $\chi_n = \mathbb{1}_{\{1,2,...,,n\}}$ in different topologies

Question

Let $X = l^2(\mathbb{N})$ and define $\chi_n = \mathbb{1}_{\{1,,2,...,n\}}$ and note that $ X = (l^\infty(\mathbb{N}))^*$. They are asking to show whether $(\chi_n)_n$ is convergent in the weak* topology and if so, give its limit. They then also ask if it is convergent in the weak topology. However, I don't really understand how these $\chi_n$ are defined or what they actually mean.