Sentence examples for be a dense subspace from inspiring English sources

Let X be a real reflexive separable Banach space with dual space X ∗ and let L be a dense subspace of X.

Theorem 4.1 Let Ω be a bounded open set in X with 0 ∈ Ω and L be a dense subspace of X.

Theorem 2.1 Let Ω be a bounded open set in X with 0 ∈ Ω and let L be a dense subspace of X. Suppose that T : D ( T ) ⊂ X → 2 X ∗ is a multi-valued operator and C : D ( C ) ⊂ X → X ∗ is a single-valued operator with L ⊂ D ( C ) such that.

Let L be a dense subspace of X and let F ( L ) denote the class of all finite-dimensional subspaces of L. Let { F n } be a sequence in the class F ( L ) such that for each n ∈ N F n ⊂ F n + 1, dim F n = n, and ⋃ n ∈ N F n ¯ = X.

In addition, the operators A 1 and A 2 given at the end of Section 2 are positively homogeneous of degree p − 1 on X = W 0 1, p ( G ). Theorem 3.1 Let L be a dense subspace of X and let λ, γ ∈ [ 1, ∞ ) be given. Suppose that T : D ( T ) = L → X ∗ is an operator and C : D ( C ) ⊂ X → X ∗ is an operator with L ⊂ D ( C ) and C ( 0 ) = 0 such that.

A set D ⊆ L p ( I, X ) is said to be 'decomposable' if for every g 1, g 2 ∈ D and for every J ⊆ I measurable, we have χ J g 1 + χ J c g 2 ∈ D. Let H be a real separable Hilbert space, V be a dense subspace of H having structure of a reflexive Banach space, with the continuous embedding V → H → V ∗, where V ∗ is the topological dual space of V.

An unbounded operator T on a Hilbert space H is defined as a linear operator whose domain D(T) is a linear subspace of H. Often the domain D(T) is a dense subspace of H, in which case T is known as a densely defined operator.

Then (D(L)) is a dense subspace of H and L is a self-adjoint operator.

Assume that is a dense subspace in and the injection of into is continuous.

Obviously, (l_{0}) is a dense subspace of (l^{p}) with (1leqslant p

Let H and V be two real separable Hilbert spaces such that V is a dense subspace of H.