Sentence examples for be a weakly closed from inspiring English sources

We obtain an analogue of Lomonosov′s Theorem for operator semigroups which are transitive on the unit sphere of H. Let A be a weakly closed unital subalgebra of B(H), and let A1 be its unit ball.

Lemma 2.11 Let X be a Banach space, let F ⊂ X be a weakly closed subset.

Lemma 2.12 Let X be a Banach space, F ⊂ X be a weakly closed subset.

Definition 2.9 Let X be a Banach space, F ⊂ X be a weakly closed subset.

Then we have ∫ 0 1 V ( u ) d t → − ∞, ∀ u n ⇀ u ∈ ∂ Λ 0. Lemma 2.3 Let X be a Banach space, and let E ⊂ X be a weakly closed subset.

Let be a weakly closed mapping where is the closed ball with center and radius.

Let X be a Banach space, F ⊂ X be a closed (weakly closed) subset, let δ ( x 1, x 2 ) be the geodesic distance between two points x 1 and x 2 in X, δ ( x, F ) be the geodesic distance between x and the set F. Suppose that Φ defined on X is Gateaux-differentiable and lower semi-continuous (or weakly lower semi-continuous) and assume Φ | F restricted on F is bounded from below.

Let X be a Banach space with E-L measure of nonconvexity μ, and let C be a nonempty, weakly closed, and bounded subset of X.

Let K be a nonempty weakly closed subset of H.

Secondly, we prove that if H is a real Hilbert space and K is a nonempty weakly closed subset of H, T : K ⊆ H → P ( H ) is a multivalued mapping from K into the family of all nonempty proximinal subsets of H. Suppose that P T is a k-strictly pseudocontractive-type mapping.

We next show that implies Suppose that is uniformly convex and is bounded and let and for all Then it is clear that and By of Lemma 2.4, we know that Since is reflexive and is weakly closed, there exists a subsequence of such that Let (3.15).