A theory of light transmission through a quantum well (QW) in a magnetic field perpendicular to the QW plane is developed. The light wave length is supposed comparable with the QW width. The formulas for reflection, absorption and transmission take into account the spatial dispersion of the light monochromatic wave and a difference of the refraction indexes of the QW and barrier. We suppose a normal light incidence on the QW plane and consider only one excited energy level. These two factors influence mostly light reflection, since an additional reflection from the QW borders appears to the reflection due to interband transitions in the QW. The most radical changes in reflection appear when a radiative broadening of the excited energy level is small in comparison to a nonradiative broadening. Our theory is limited by the condition of existence of size-quantized energy levels which is satisfied for quite narrow QW's.