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 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.
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.
An algebra norm on ℬ is a map ∥ ⋅ ∥ : B → [ 0, ∞ ) such that ( B, ∥ ⋅ ∥ ) is a normed space, and, further ∥ a b ∥ ≤ ∥ a ∥ ∥ b ∥. for any a, b ∈ B. The normed algebra ( B, ∥ ⋅ ∥ ) is a Banach algebra if ∥ ⋅ ∥ is a complete norm.
Form the direct sum (C([a,b],H oplus F_{c}) and give it the complete norm |y+z1_{[a,c)}|=sup_{sin[a,b]}biglvert y(s bigrvert +|z|, quad yin Cbigl([a,b],Hbigr), zin H. Now let us give two basic lemmas that will be used in the following sections (see Lemma 2.1 and Lemma 2.2 in [26]).
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}}.
