Topology on the space of matrices

Question

The vector space of $n\times n$ real matrices is isomorphic as a vector space to $\mathbb R^{n\times n}$. Does it follow that it is "the same" as the topological space $\mathbb R^{n\times n}$ with the standard topology (e.g. the box topology, which is the same as the product topology, which is the same as the topology generated by balls)? (What exactly does this statement (i.e. "the same") mean?) If I know an open set in $\mathbb R^{n\times n}$ (say $(a,b)^{\times n^2})$, how do I find the corresponding open set in the space of matrices?

Answer 1

The vector space of $n\times n$ matrices is essentially just $\mathbf{R}^{n\times n}$ with the coordinates written in a square instead of a straight line, so it's obvious that the two are isomorphic. All norms on finite dimensional vector spaces are equivalent, which means they induce they same topology.

Two topological vector spaces are isomorphic ("the same") if they are isomorphic as vector spaces, and the isomorphism is also a homeomorphism. Since all norms are equivalent, functions continuous with respect to one norm will be continuous in all norms. We can use the same norm, say the Euclidean norm, on both vector spaces, and the required isomorphism is essentially the identity map. (To map from $\mathbf{R}^{n\times n}$ to the space of matrices, for example, just arrange the coordinates, in order, in an $n\times n$ square.)