This study (analogous to luminosity-function determinations for galaxies) sets the resolution demand for the UV telescope by asking to study the sphere of influence of the central objects in AGNs. Four milliarcsecond angular resolution meets this need, for the largest black holes, anywhere in the Universe. The study demands full aperture, not interferometry, because of the large number of objects that need to be measured. Lower mass objects have smaller spheres of influence; the telescope scale adequate to do the more distant, high mass objects, is adequate for the low mass black holes, at low z. A larger volume is needed, containing distant AGNs, to obtain a large sample of massive objects, but for low masses, lower volumes (redshift) are sufficient, as the low mass holes have a higher number per cubic megaparsec.