continuous functions of finite dimension banach spaces in infinite dimension banach spaces

Question

I have the following problem but I am not sure how to proceed. I that could suppose that there is at least one continuous function, but afterwards I don't know how to continue.

Prove that there is no continuous function of a finite dimension Banach X space over an infinite dimension Banach Y space.

I would appreciate any suggestions.

Answer 1

Consider the balls $nB_X$ in $X$, centred at $0$, radius $n$. Note that these sets are compact and union to give $X$. If we have a continuous surjective function $f : X \to Y$, then $f(nB_X)$ are compact and union to give $Y$. But according to my answer here, this would imply $Y$ is finite-dimensional.