24.11. SOME EMBEDDING THEOREMS 691

1. A ⊆C (K;W ) and A is equicontinuous into W

2. A is pre-compact in C (K;W ) . This means that A is compact in C (K;W ).

Proof: Let C ≡ sup f∈A ∥ f∥∞,V < ∞ . Let ε > 0 be given. Then from Lemma 24.11.1,

∥ f (x)− f (y)∥W ≤ε

5C∥ f (x)− f (y)∥V +Cε ∥ f (x)− f (y)∥U ≤

2ε

5+Cε ∥ f (x)− f (y)∥U

By equicontinuity in C (K,U) , there exists a δ > 0 such that if d (x,y) < δ , then for allf ∈ A , Cε ∥ f (x)− f (y)∥U < 2ε

5 . Thus if d (x,y) < δ , then ∥ f (x)− f (y)∥W < ε for allf ∈A .

It remains to verify that A is pre-compact in C (K;W ) . Since this space of continuousfunctions is complete, it suffices to verify that for all ε > 0, A has an ε net. Supposethen that for some ε > 0 there is no ε net. Thus there is an infinite sequence { fn} forwhich ∥ fn− fm∥∞,W ≥ ε whenever m ̸= n. There exists δ > 0 such that if d (x,y) < δ ,

then for all fn, ∥ fn (x)− fn (y)∥W < ε

5 .Let {xk}pk=1 be a δ/2 net for K. This is where we

use K is compact. By compactness of the embedding of V into W, there exists a furthersubsequence, still called { fn} such that each { fn (xk)}∞

n=1 converges, this for each xk inthat δ/2 net. Thus there is a single N such that if n > N, then for all m,n > N, andk ≤ p,∥ fn (xk)− fm (xk)∥W < ε

5 . Now letting x ∈ K be arbitrary, it is in B(xk,δ/2) forsome xk. Therefore, for n,m larger than N,

∥ fn (x)− fm (x)∥W ≤ ∥ fn (x)− fn (xk)∥W +∥ fn (xk)− fm (xk)∥W +∥ fm (xk)− fm (x)∥

<ε

5+

ε

5+

ε

5=

3ε

5Taking the maximum for all x, for m,n>N,∥ fn− fm∥W,∞ ≤ 3ε

5 < ε contrary to the assump-tion that every pair is further apart than ε . Thus A is totally bounded so its closure wouldalso be totally bounded and complete. In other words, A is pre-compact in C (K;W ). ■

In the following theorem about compact subsets of an Lp space, the measure will beLebesgue measure. It depends on the above version of the Ascoli Arzela theorem. Firstnote the following which I will use when convenient. For a,b ≥ 0, and p ≥ 1, then byconvexity of φ (t) = t p for t ≥ 0, (a+b)p ≤ 2p−1 (ap +bp). Also, for such p,(a+b)1/p ≤a1/p+b1/p. Usually the thing of interest in this theorem is the case where V =W =U =R.However, the more general version to be presented is interesting I think. Of course closedand bounded sets are compact in R so the usual case works as a special case of what isabout to be presented.

Theorem 24.11.4 Let V ⊆W ⊆ U where these are Banach spaces such that theinjection map of V into W is compact and the injection map of W into U is continuous. LetΩ be an open set in Rm and let A be a bounded subset of Lp (Ω;V ) and suppose that forall ε > 0, there exist a δ > 0 such that if |h|< δ , then for ũ denoting the zero extension ofu off Ω, ∫

Rm∥ũ(x+h)− ũ(x)∥p

U dx < εp (24.42)

Suppose also that for each ε > 0 there exists an open set, Gε ⊆ Ω such that Gε ⊆ Ω iscompact and for all u ∈A , ∫

Ω\Gε

∥u(x)∥pW dx < ε

p (24.43)

Then A is precompact in Lp (Rn;W ).