On the closure of the sum of closed subspaces

Abstract

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.2000 Mathematics Subject Classification. 46C05, 90C20.

Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

“…However, it was shown in [8] that P (F ) is not closed, so they cannot possibly be equal. More directly, we exhibit an element of ∞ j =1 P (F j ) which does not belong to P (F ).…”

“…Here, we are also interested in finding conditions for P (F ) to be weakly closed and for c(K, N) to be strictly less than 1. In [8,Theorem 4.1], we established the equivalence of (i) and (ii) in the following result. (See also [2,Theorem 9.35]. )…”

“…In [8], we were interested in finding conditions for P (F ) to be norm closed. Here, we are also interested in finding conditions for P (F ) to be weakly closed and for c(K, N) to be strictly less than 1.…”

Products of square-zero operators

“…Let P denote the orthogonal projection onto H 1 . Since N + im T is a closed space, the subspace H 12 = P (N ) is also closed by [6,Theorem 2.1]. Since N and H 2 have trivial intersection, P is injective on N .…”

The Chiral Anomaly of the Free Fermion in Functorial Field Theory

“…To see these statements imply that L 02 is a Lagrangian, we use Thm. 2.1 in [19], which states that for two closed subspaces X, Y ⊆ W ,…”