In　this　paper,　we　introduce　the　concept　of　biholomorphic　convex　mapping　of　order　a　on　the　unit　ball　B　in　a　complex　Hilbert　space　X.　Then　we　provide　some　sufficient　conditions　that　a　locally　biholomorphic　mapping　f　is　a　biholomorphic　convex　mapping　of　order　a;　we　give　an　Alexander　type　theorem　between　the　subclass　of　convex　mappings　and　the　subclass　of　starlike　mappings　on　B　in　a　Hilbert　space.　We　also　study　the　order　of　starlikeness　of　biholomorphic　convex　mappings　of　order　a　on　B　in　a　Hilbert　space.　Finally,　we　construct　some　examples　of　biholomorphic　convex　mappings　of　order　a　on　B　in　a　Hilbert　space　by　a　linear　operator.　(C)　2013　Elsevier　Inc.　All　rights　reserved.