Sentence examples for as a closed subspace from inspiring English sources

It is easy to see that (V_{t}(Omega)) is reflexive and separable as a closed subspace of (W_{0}^{1,p^(Omega )cap L^{2}(Omega )).

In particular, we set mathfrak{Lip}_{0} X,mathbb{K}):= X^{sharp} and call (X^{sharp}) the pseudo-dual space of X; this space contains the usual dual space (X^{ast}) as a closed subspace.

(iii) A metric space A is an approximative absolute neighborhood retract ((Ain AANR) for short) if and only if A is an approximative neighborhood retract of any metric space ((X,d)) in which it is imbedded as a closed subspace; i.e., for any (epsilon>0), there exists an open neighborhood V of A in X and a continuous mapping (r:Vrightarrow A) such that (d(r(a),a)

Each of the Hi is included as a closed subspace in the direct sum of all of the Hi.

We introduce a natural Banach algebra, which is a closed subspace of ⟨A∗A⟩∗ but equipped with a distinct multiplication.

Let X be a Banach space and let Y be a closed subspace of a Banach space Z.

Let Y be a closed subspace of Lp, where μ is an arbitrary measure and 1 < p < ∞.

Let X be a closed subspace of a Banach space Y and J be the inclusion map.

Let be a closed subspace of.

Lemma 4.9. is a closed subspace of.

Let (Ysubset X) be a closed subspace.