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how can i prove that the multiplication map: $$\operatorname{mult}: M_n(\mathbb{K}) \times M_n(\mathbb{K}) \to M_n(\mathbb{K});\operatorname{mult}(A,B) \mapsto AB $$ and the addition-map: $$\operatorname{add}:M_n(\mathbb{K}) \times M_n(\mathbb{K}) \to M_n(\mathbb{K}); \operatorname{add}(A,B) \mapsto A+B $$ are continuous on the product topology on $M_n(\mathbb{K}) \times M_n(\mathbb{K})$?
We consider the topology induced by the norm $\Vert A \Vert=\max\{\vert Ax \vert: x \in \mathbb{K}^n, \vert x \vert =1\}$.
Thank you!

Michael Hardy
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leon
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    They are both componentwise polynomial function. A function is continuous iff it is continuous in every component. – Blumer Nov 07 '17 at 18:26
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    If $\Bbb K$ is $\Bbb R$ or $\Bbb C$ then all these spaces are finite-dimensional real vector spaces. All Hausdorff topological vector space topologies on a finite-dimensional real vector space are equivalent, so it doesn't really matter which norm one is using. – Angina Seng Nov 07 '17 at 18:43

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Let $A_{n}\rightarrow A$ and $B_{n}\rightarrow B$ in the norm-topology, then $\|A_{n}B_{n}-AB\|\leq\|(B_{n}-B)A_{n}\|+\|A_{n}B-AB\|\leq\|A_{n}\|\|B_{n}-B\|+\|A_{n}-A\|\|B\|$, we know that $\sup_{n}\|A_{n}\|<\infty$.

On the other hand, $\|(A_{n}+B_{n})-(A+B)\|\leq\|A_{n}-A\|+\|B_{n}-B\|$.

user284331
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