  In this paper, we consider the quadratic programming problems under finitely
many convex quadratic constraints in Hilbert spaces. By using the Legendre
property of quadratic forms or the compactness of operators in the
presentations of quadratic forms, we establish some sufficient conditions for
the solution existence of the considered problems. As special cases, we obtain
some existence solution results for the quadratic programming problems under
linear constraints in Hilbert spaces.
