International Journal of Mathematics and Mathematical Sciences
Volume 26 (2001), Issue 5, Pages 257-267

On the closure of the sum of closed subspaces

Irwin E. Schochetman,1 Robert L. Smith,2 and Sze-Kai Tsui1

1Mathematics and Statistics, Oakland University, Rochester 48309, MI, USA
2Industrial and Operations Engineering, The University of Michigan, Ann Arbor 48109, MI, USA

Received 30 May 2000

Copyright © 2001 Irwin E. Schochetman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We give necessary and sufficient conditions for the sum of closed subspaces of a Hilbert space to be closed. Specifically, we show that the sum will be closed if and only if the angle between the subspaces is not zero, or if and only if the projection of either space into the orthogonal complement of the other is closed. We also give sufficient conditions for the sum to be closed in terms of the relevant orthogonal projections. As a consequence, we obtain sufficient conditions for the existence of an optimal solution to an abstract quadratic programming problem in terms of the kernels of the cost and constraint operators.