Sentence examples for be a subspace in from inspiring English sources

Let T be a subspace in X 2 and J be a conjugation operator.

Let T be a subspace in X 2. Then T ∗ is a closed subspace in X 2, T ∗ = ( T ¯ ) ∗, and T ∗ ∗ = T ¯, where T ¯ is the closure of T. Definition 2.2 (see [[19], p.114] or [3]).

Let T be a subspace in X 2. R ( T − λ I ) ⊥ is called the defect space of T and λ, and dim ( R ( T − λ I ) ⊥ ) is called the defect index of T and λ.

Let T and S be a subspace in X 2. The product of T and S is defined by T S = { ( x, g ) ∈ X 2 : ( x, f ) ∈ S, ( f, g ) ∈ T }.

Let T be a subspace in X 2. (1) Its adjoint, T ∗, is defined by T ∗ = { ( y, g ) ∈ X 2 : 〈 f, y 〉 = 〈 x, g 〉  for all  ( x, f ) ∈ T }.   (2) T is said to be a Hermitian subspace if T ⊂ T ∗.   (3) T is said to be a self-adjoint subspace if T = T ∗.  .

Let T be a subspace in X 2. The set Γ ( T ) : = { λ ∈ C :  there exists  c > 0  such that ∥ f − λ x ∥ ≥ c ∥ x ∥  for all  ( x, f ) ∈ T }. is called to be the regularity field of T. First, we give a result for the regularity field of Γ ( H 0 . Lemma 7.1 Assume that a is finite.

It is clear that E 1 is the ideal lattice on G and is a subspace in the space dual to E. Ideal lattices generalize many important spaces e.g. spaces L p ( G ), 1 ⩽ p ⩽ ∞, the Orlicz spaces [26], the Lorentz spaces [18], the Marcinkiewicz spaces [18], etc.

Let E be a subspace of all (x in X) for which (C t) x) is a once continuously differentiable function of t.

This is a subspace of functions in (M_{p,lambda}({mathbb{R}}^{n})), which satisfy the condition begin{aligned} lim_{rrightarrow0}sup_{xin {mathbb{R}}^{n}, 0< t< r} t^{-frac{lambda }{p}} Vert f Vert _{L_{p}(B x,t))}=0.

Let X 2 be the product space X × X with the following induced inner product, denoted by 〈 ⋅, ⋅ 〉 without any confusion: 〈 ( x, f ), ( y, g ) 〉 = 〈 x, y 〉 + 〈 f, g 〉 for all  ( x, f ), ( y, g ) ∈ X 2. Let T be a linear subspace in X 2. For briefness, a linear subspace is only called a subspace.

The closure of a subspace can be completely characterized in terms of the orthogonal complement: If V is a subspace of H, then the closure of V is equal to V^{\bot\bot}.