下面這一段英文 我看不太懂 所以想請大家講解一下~
Several important properties of the space W^{m,p}(Ω) can be easily obtained
by regarding W^{m,p}(Ω) as a closed subspace of L^p space on a union of
disjoint copies of Ω.
Given integer n≧1 and m≧0, let N≡N(n,m) be the number of multi-indices
α= ( α_1, ... , α_n ) such that |α|≦m. For each such multi-index α, let
Ω_α be a copy of Ω in a different copy of R^n, so that N domains Ω_α
are de facto mutually disjoint. Let Ω^(m) be the union of these N domains;
Ω^(m) = ∪ Ω_α. Given a function u in W^{m,p}(Ω), let U be the
|α|≦m
function on Ω^(m) that coincides with (D^α)u on Ω_α. It is easy to check
that the map P taking u to U is an isometry from W^{m,p}(Ω) into L^p(Ω^(m)).
Since W^{m,p}(Ω) is complete, the range W of the isometry P is a closed
subspace of $ L^p(Ω^(m)). It follows that W is separable if 1 ≦ p < ∞,
and is uniformly convex and reflexive if 1 < p < ∞. The same conclusions must
therefore hold for W^{m,p}(Ω) = P^{-1}(W).
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