Sentence examples for a basis of v from inspiring English sources

This said, a basis of (V otimes bar{V}) is given by the set ({ g otimes bar{chi }}).

Let us fix some k and assume that the dimension of V k is r and ϕ 1, …, ϕ r is a basis of V k.

Pick (x_i in V_i - 0); then ((x_i)_{iin mathbb {I}}) is a basis of V, and (c(x_iotimes x_j)=q_{ij}, x_jotimes x_i), (i,jin mathbb {I}), where (q_{ij} in mathbb {k}^{times }).

We consider R as the path algebra of the quiver with one vertex and one loop; in this way, the R -modules of dimension n are just pairs ( V, ϕ ), where V is a k -space of dimension n and ϕ : V → V an endomorphism of V, or, after choosing a basis of V, we just deal with ( n × n ) -matrices with coefficients in k.

If (partial _{f} (x) =0) for all f in a basis of (V^*), then (x in {mathcal {J}}^n V)).

If (widetilde{partial }_{f} (x) =0) for all f in a basis of (V^*), then (x in {mathcal {J}}^n V)).  .

If (widetilde{partial }_{f} (x) =0) for all f in a basis of (V^*), then (x in {mathcal {J}}^n V)).

([78]) With the above notions, a linear basis of V consists of the elements a 1 ( n 1 ) a 2 ( n 2 ) ⋯ a k ( n k ), a i ∈ B, n i ∈ Z satisfying the condition in Corollary 3 and n k < 0. Clearly, as k ⟨ X ⟩ -modules, we have U V = U ( U ( L ) / U ( L ) L + ) = Mod k ⟨ X ⟩ ⟨ I | S X ∗ I, a ( n ) I, n ≥ 0 ⟩ = k ⟨ X ⟩ ⟨ I | S ′ ⟩, where S ′ = { S X ∗ I, a ( n ) I, n ≥ 0 }.

Definition 1 A multiresolution analysis of L 2 ( R ) means a sequence of closed linear subspaces V j of L 2 ( R ) which satisfies (i) V j ⊂ V j + 1, j ∈ Z,   (ii) f ( x ) ∈ V j if and only if f ( 2 x ) ∈ V j + 1,   (iii) ⋃ j ∈ Z V j ¯ = L 2 ( R ) and ⋂ j ∈ Z V j = { 0 },   (iv) there exists a function ϕ ∈ L 2 ( R ) such that { ϕ ( x − k ), k ∈ Z } forms a Riesz basis of  V 0.  .

V j ⊂ V j + 1, j ∈ Z, f ( x ) ∈ V j if and only if f ( 2 x ) ∈ V j + 1, ⋃ j ∈ Z V j ¯ = L 2 ( R ) and ⋂ j ∈ Z V j = { 0 }, there exists a function ϕ ∈ L 2 ( R ) such that { ϕ ( x − k ), k ∈ Z } forms a Riesz basis of  V 0. The function ϕ in Definition 1 is said to be a scaling function, if it satisfies ϕ ( x ) = ∑ k a k ϕ ( 2 x − k ) (1.2). for some sequence { a k } ∈ ℓ 2 ( Z ).

As before, if ({ e_i }) is an orthonormal basis of V let (f_i := c e_i) denote the dual basis of (V^*).