Let $F/E$ be a finite extension of local fields, and $N = N_{F/E} : F^* \to E^*$ be the norm map (it is continuos). And consider an axiom:
$N$ has closed image and compact kernel.
In the book, the author says that this condition is equivalent to that $N$ is proper. But I can't show it. Please help.
P.S. This condition holds in this case. I use the word "axiom" because I use the explicit terminology for simplification (the author uses class field formation). So please show the equivalence using only locally compactness, countability at infinity of $E, F$ and continuity of $N$.