Show that a function is continuous on a normed space

Question

So I have learned topology and we haven't touched normed space there. However, from my linear algebra class I know that every normed space can naturally induce a metric, so I wonder that, to show $f:X\rightarrow X$ is continuous on the normed space $X$, can we treat $X$ as a metric space and show continuity using $\epsilon$-balls argument (because that's what I'm currently comfortable with).

Anyway, what do we actually mean that a function is continuous when its domain and codomain are some normed space. I'm not very sure about this. Could anyone present a definition and reference?

Answer 1

You you can do that, knowing that the distance is defined by $d(x, y) = \Vert x - y \Vert$.

Answer 2

Yes, you can.

However, $\epsilon$-$\delta$ proofs are a bit cumbersome. It's almost always easier to use the topological definition when you can.