In　this　paper,　we　give　a　necessary　and　sufficient　condition　that　a　locally　biholomorphic　mapping　f　on　the　unit　ball　B　in　a　complex　Hilbert　space　X　is　a　biholomorphic　convex　mapping,　which　improves　some　results　of　Hamada　and　Kohr　and　solves　the　problem　which　is　posed　by　Graham　and　Kohr.　From　this,　we　derive　some　sufficient　conditions　for　biholomorphic　convex　mapping.　We　also　introduce　a　linear　operator　in　purpose　to　construct　some　concrete　examples　of　biholomorphic　convex　mappings　on　B　in　Hilbert　spaces.　Moreover,　we　give　some　examples　of　biholomorphic　convex　mappings　on　B　in　Hilbert　spaces.　©　2005　Elsevier　Inc.　All　rights　reserved.