Matrix sequence limit convergence

Question

Let $A_k$ be a sequence of matrices such that $\lim A_kx=Ax$ for all $x\in \mathbb{R}^n$. Then I need to prove that $A_k \rightarrow A$ with respect to any norm.

Answer 1

The key result here is that any two norms on the same finite dimensional space are equivalent. Hence it is sufficient to show the result for a specific (and convenient) norm.

Define $\|A\|_* = \sum_i \|A e_i\|$, where $e_i$ are the standard basis vectors. It is easy to check that this is a norm, and since $\|A-A_k\|_* = \sum_k \|(A-A_k)e_i\|$, we see that $\|A-A_k\|_* \to 0$.

Answer 2

Since for every basic vector with have the mentioned convergence, we have that the entries on $A_k=(a^k_{i,j})$ must converge to the entries on $A = (a_{i,j})$. Take $|A| = \max|a_{i,j}|$.
Then there exists $K>0$ such that for all $k>K$, $|a^k_{i,j}-a_{i,j}|<\epsilon $ for all $i,j$ (since there are finitely many). So, for $k>K$, we have $$|A_k-A| = \max |a^k_{i,j}-a_{i,j}| <\epsilon$$