1266 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

It follows Dα j ( f1) = f j and so Λ(Xm,p (U)) is closed as claimed. This is clearly also asubspace of Lp (U)w and so it follows that Λ(Xm,p (U)) is a reflexive Banach space. This isbecause Lp (U)w, being the product of reflexive Banach spaces, is reflexive and any closedsubspace of a reflexive Banach space is reflexive. Now Λ is an isometry of Xm,p (U) andΛ(Xm,p (U)) which shows that Xm,p (U) is a reflexive Banach space. Finally, W m,p (U) isa closed subspace of the reflexive Banach space, Xm,p (U) and so it is also reflexive. Tosee Xm,p (U) is separable, note that Lp (U)w is separable because it is the finite product ofthe separable hence completely separable metric space, Lp (U) and Λ(Xm,p (U)) is a subsetof Lp (U)w . Therefore, Λ(Xm,p (U)) is separable and since Λ is an isometry, it followsXm,p (U) is separable also. Now W m,p (U) must also be separable because it is a subset ofXm,p (U) .

The following theorem is obvious but is worth noting because it says that if a functionhas a weak derivative in Lp (U) on a large open set, U then the restriction of this weakderivative is also the weak derivative for any smaller open set.

Theorem 37.0.4 Suppose U is an open set and U0 ⊆U is another open set. Suppose alsoDα u ∈ Lp (U) . Then for all ψ ∈C∞

c (U0) ,∫U0

(Dα u)ψdx = (−1)|α|∫

U0

u(Dαψ) .

The following theorem is a fundamental approximation result for functions in Xm,p (U) .

Theorem 37.0.5 Let U be an open set and let U0 be an open subset of U with the propertythat dist

(U0,UC

)> 0. Then if u ∈ Xm,p (U) and ũ denotes the zero extention of u off U,

liml→∞

||ũ∗φ l−u||Xm,p(U0)= 0.

Proof: Always assume l is large enough that 1/l < dist(U0,UC

). Thus for x ∈U0,

ũ∗φ l (x) =∫

B(0, 1l )

u(x−y)φ l (y)dy. (37.0.1)

The theorem is proved if it can be shown that Dα (ũ∗φ l)→ Dα u in Lp (U0) . Let ψ ∈C∞

c (U0)

Dα (ũ∗φ l)(ψ) ≡ (−1)|α|∫

U0

(ũ∗φ l)(Dα

ψ)dx

= (−1)|α|∫

U0

∫ũ(y)φ l (x−y)(Dα

ψ)(x)dydx

= (−1)|α|∫

Uu(y)

∫U0

φ l (x−y)(Dαψ)(x)dxdy.