I have been give this question. I am just a bit confused where to start?
Prove that $g : \mathrm{GL}(n,\mathbb R)\times \mathrm{GL}(n,\mathbb R)\to \mathrm{GL}(n,\mathbb R)$ given by $g(x, y) := xy$ (matrix multiplication) is continuous, where we take the relative topology from $\mathbb R^{n^2}\times ×\mathbb R^{n^2}\to \mathbb R^{n^2}$ on $\mathrm{GL}(n,\mathbb R)\times \mathrm{GL}(n,\mathbb R)$.
I just don't know where to take into account the relative topology?
Hint. For each matrix $x=(x_{ij})\in GL(n,\Bbb R)$ define it's norm $\|x\|=\max |x_{ij}|$. A metric $d(x,y)=\|x-y\|$ for any matrices $x,y\in GL(n,\Bbb R)$ induces on $GL(n,\Bbb R)$ the topology inherited from $\Bbb R^{n^2}$. Now if $x,x',y, y'\in GL(n,\Bbb r)$ are matrices such that $\|x'-x\|<\varepsilon$ and $\|y'-y\|<\varepsilon$ then $\|x'y'-xy\|<n(\|y\|+\|x'\|)\varepsilon$. Indeed, for any indices $i,j$ we have
$$(xy)_{ij}=\sum_{k=1}^n x_{ik} y_{kj},$$
$$(x’y’)_{ij}=\sum_{k=1}^n x_{ik}’ y_{kj}’.$$
So $$(xy)_{ij}-(x’y’)_{ij}=\sum_{k=1}^n x_{ik}y_{kj}- x_{ik}’y_{kj}’\le$$
$$\sum_{k=1}^n (x_{ik}-x_{ik}’)y_{kj}+ x_{ik}’(y_{kj}-y_{kj}’)\le $$
$$n(\|x-x’\|\|y\|+\|y-y’\|\|x'\|).$$