Continuity of matrix product with respect to matrix norm? I'm trying to teach myself about ordinary differential equations with an old script and I'm struggling with this problem: 
Show that the matrix product is continuous with respect to
the matrix norm. That is, if $A_j → A$ and $B_j → B$ we have $A_j\cdot B_j → AB$.
My problem is that I don't even understand what the limit of $A_j\cdot B_j$ has to do with the matrix norm. 
Any help would be greatly appreciated!
 A: Well you need a notion of norm in order to speak of convergence,i.e. 
$$A_jB_j\longrightarrow AB\mathrm{\ \ iff\ \ }\forall\epsilon>0,\exists N>0\mathrm{\ s.t.\ }j>N\Rightarrow\Vert AB-A_jB_j\Vert<\epsilon$$
Anyway here you don't need to go back to the definition to prove that the product is continuous. You basically need properties which are easy to prove :


*

*The product on real/complex numbers is continuous

*The sum on real/complex numbers is continuous

*The projections ($f(x)=x_i$, $\mathbb R^n\rightarrow \mathbb R$) on any coordinates of a vector are continuous

*The "suspension" ($f_i(x)=(0,...,x,...,0)$, $\mathbb R\rightarrow \mathbb R^n$), that is, the function that associates a real number with a vector containing only zeros but $x$ at the i-th coordinate is also continuous

*Most important : if $f,g$ are two (composable) continuous functions, then their composition is also continuous.


With that being said, notice that the matrix product is but a composition of all these different functions which are all continuous.
Hence it's continuous.
A: In general: continuity can be defined only for functions between topological spaces.
It seems that you refer to a definition of continuity in terms of limits, and this is usually done in a metric space (that is also a topological space) (see here) .
A vector space of matrices as, for example, $M(n, \mathbb{R})$ (that is a vector space) becomes a metric space if we define  a norm that induce a distance in the canonical way:
$$
d(A,B)=||A-B||
$$
So, if we have a norm, we have metric space and we can define convergence and continuity.
