Upper Bounds for the Distance to Finite-Dimensional Subspaces in Inner   Product Spaces
  We establish upper bounds for the distance to finite-dimensional subspaces in inner product spaces and improve some generalisations of Bessel's inequality obtained by Boas, Bellman and Bombieri. Refinements of the Hadamard inequality for Gram determinants are also given. 
Sever Silvestru Dragomir