Sentence examples for complete norm from inspiring English sources

If the metric of an FK space is given by a complete norm, then we say that X is a BK space.

Let X = C ( J, R n ) be the Banach space of all continuous functions from J into R n with the norm ∥ x ∥ = sup t ∈ J | x ( t ) |, where | ⋅ | denotes a suitable complete norm on R n.

end{aligned}Then, (W^{k,1}(Omega )) is equipped with the complete norm begin{aligned} Vert fVert _{k,1}=max _{0le |alpha |le k}Vert D^{alpha }fVert _{L^1(Omega )}.

If the trivial representation of G is not weakly contained in the left regular representation of G on L02(G) and X is either Lp G) for 1<p⩽∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||p or ||·||∞ respectively.

Let A be a Banach function space and let M be a family of multipliers on A. We provide conditions on M so that the original topology of A is the only complete norm topology on A making all of the maps from M continuous.

If 1<p⩽∞ and every left invariant linear functional on Lp G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on Lp G) that makes translations from (Lp(G),|·|) into itself continuous and that makes the map t↦Lt from G into L Lp(G),|·|) bounded is equivalent to ||·||p.

The extension generates a set of world-norm pairs by pairing each possible world with each possible set of consistent and complete norms.

Then this is a complete normed space with a norm (2.7).

Then this is a complete normed space with a norm Vert xVert _{W}=Bigl[sum vert mu_{k} vert h^{2}_{k}Bigr]^{frac{1}{2}}.

Then this is a complete normed space with a norm ∥ u ∥ = [ ∑ | Λ k | h k 2 ] 1 2. Since λ k → + ∞ and c is fixed, we have Λ k → ∞ and (i) Δ 2 u + c Δ u ∈ H implies u ∈ H,   (ii) ∥ u ∥ ≥ C ∥ u ∥ L 2 for some C > 0,   (iii) ∥ u ∥ L 2 = 0 if and only if ∥ u ∥ = 0,  .

This proves that N ( λ, p ) is a complete normed space under the Luxemburg norm.