Add to ORKGAbstract. 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
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We give an affirmative answer to the invariant subspace problem for densely defined closed operators...
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We give necessary and sufficient conditions for the sum of closed subspaces of a Hilbert space to be...
Abstract In this article, a new characterization of the closedness of the sum of two closed subspace...
Suppose one has two closed linear subspaces. Is their sum also closed? It turns out that it is not n...
AbstractSome results are presented giving conditions under which the algebraic sum of two closed lin...
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Let A be a linear space of operators on a Hilbert space H, x a vector in H,and Ax the subspace of H ...
AbstractWe first present a formula for the supremum cosine angle between two closed subspaces of a s...
We give a new proof of a characterization of the closeness of the range of a continuous linear ope...
AbstractSuppose T and S are bounded adjointable operators with close range between Hilbert C∗-module...
This paper deals with the characterization of the sums of compact convex sets with linear subspaces...
A multivalued linear projection operator P defined on linear space X is a multivalued linear opera...
AbstractA formula is given for the orthogonal complement of any vector subspace of l2. Countably inf...
We give an affirmative answer to the invariant subspace problem for densely defined closed operators...
AbstractIn this paper we find a necessary and sufficient condition for two closed subspaces, X and Y...
Sum and intersection of linear subspaces in a vector space over a field are fundamental operations i...