  We show that some Gieseker stable sheaves on a projective K3 surface $X$ are
stable with respect to a stability condition of Bridgeland on the derived
category of $X$ if the stability condition is in explicit subsets of the space
of stability conditions depending on the sheaves. Furthermore we shall give two
applications of the result. As a part of these applications, we show that the
fine moduli space of Gieseker stable torsion free sheaves on a K3 surface with
Picard number one is the moduli space of $\mu$-stable locally free sheaves if
the rank of the sheaves is not a square number.
