I know that the directivity of the monopole is twice the directivity of the dipole because the transmitted power in case of the monopole is distributed only across a half of the space.
But why the input impedance of the monopole is half of the impedance of the dipole? Is there some proof to that?
It's a simple symmetry argument:
from https://en.wikipedia.org/wiki/Monopole_antenna
The fields from the two arrangements in the figure are the same above the ground. The monopole has the same current but half the voltage, hence has half the impedance. For the same current it also radiates half the power as it is radiating into half the space.
