  We present K-band observations of the low-luminosity galaxies in the Coma
cluster, which are responsible for the steep upturn in the optical luminosity
function at M_R ~ -16, discovered recently. The main results of this study are
(i) The optical$-$near-infrared colours of these galaxies imply that they are
dwarf spheroidals. The median M-K colour for galaxies with -19.3 < M_K < -16.3
is 3.6 mag. (ii) The K-band luminosity function in the Coma cluster at the
faint-end is not wee constrained, because of the uncertainties due to the
field-to-field variance of the background. However, within the estimate large
errors, it is consistent with the R-band luminosity function, shifted by
$\sim3$ magnitudes. (iii) Many of the cluster dwarfs lie in a region of the B-K
vs. B-R colour-colour diagram where background galaxies are rare Local dwarf
spheroidal galaxies lie in this region too. This suggests that a better
measurement of the K-band cluster luminosity function can be made if the
field-to-field variance of the background can be measured as a function of
colour. (iv) If we assume that none of the galaxies in the region of the B-K
vs. B-R plane given in (iii) in our cluster fields are background, and that all
the cluster galaxies with $15.5 < K < 18.5$ lie in this region of the plane,
then we measure alpha = -1.41 +/- 0.35 for -19.3 < M_K < -16.3, where alpha is
the logarithmic slope of the luminosity function.
