International Journal of Mathematics and Mathematical Sciences
Volume 26 (2001), Issue 5, Pages 257-267
doi:10.1155/S0161171201005324
On the closure of the sum of closed subspaces
Irwin E. Schochetman1
, Robert L. Smith2
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
Abstract
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.