I have a sequence of $n \times n$ matrices $\{F_i\}_{i=1}^{\infty}$. Let $a_{k,ij}$ be the $i,j$ th entry of $F_k$ matrix and $a_{ij}$ be the $i,j$ th entry of $F$. Its given that $a_{k,ij} \rightarrow a_{ij}, \hspace{1mm} \forall k$ in $p \geq 2$ norm, . We may take $$||F||= \sum_{i,j=1}^n|a_{ij}|^2 $$
Here I want to discuss
What different types of convergence one can show for $F_k$ converging to $F$ (example in p norm, square norm, etc, I hope that makes sense)?