We have analysed the proper motion data of LHS catalog, to derive the faint end of the luminosity function more precise than ever before, by mean absolute method, and by making use of the reudced proper motion diagram. It is found that the relations between the mean absolute magnitude and the reduced proper motion for main sequence stars, subdwarfs, and white dwarfs are so different that the proper application of an appropriate relation to each group is much more important. The derived luminosity function shows the broad maximum peak from M B ∼ 14 to M B ∼ 17 and declines after M B ∼ 17 up to M B ∼ 22 .

Two theories, the adiabatic and the isothermal, of galaxy formation are surveyed with regard to the current observational constraints, Some advantage of non-baryonic theory compared with the usual baryonic theory is discussed.