Suppose $L(\mathbb R^n)$ denotes the set of all invertible linear transformation from $\mathbb R^n$ to itself. It is well known that it is a metric space induced by the norm $||A||=\sup_{||x||<1}||Ax||$. We also know that $A\mapsto A^{-1}$ is a continuous mapping. If we define multiplication as composition in $L(\mathbb R^n)$ , is it true that the mapping $F:L(\mathbb R^n)\times L(\mathbb R^n) \to L(\mathbb R^n)$ defined as $F(A,B)=A\circ B$ is continuous? [If it is true then it in turn will prove that $L(\mathbb R^n)$ is a topological group]
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$F$ is a bilinear map. And a bilinear map on finite dimensional spaces is always continous.