In　this　paper,　we　first　determine　the　relations　among　the　best　bounds　A　and　B　of　the　g-frame,　the　g-frame　operator　S　and　the　pre-frame　operator　Q　and　give　a　necessary　and　sufficient　condition　for　a　g-frame　with　bounds　A　and　B　in　a　complex　Hilbert　space.　We　also　introduce　the　definition　of　a　g-frame　sequence　and　obtain　a　necessary　and　sufficient　condition　for　a　g-frame　sequence　with　bounds　A　and　B　in　a　complex　Hilbert　space.　Lastly,　we　consider　the　stability　of　a　g-frame　sequence　for　a　complex　Hilbert　space　under　perturbation.　©　The　Editorial　Office　of　AMS　&Springer-Verlag　2009.