206 CHAPTER 9. WEIERSTRASS APPROXIMATION THEOREM
2. ↑ In the context of Problem 1, suppose R= Y where the usual topology is placed onR. Show f achieves its maximum and minimum on A.
3. Let V be an open set in Rn. Show there is an increasing sequence of compact sets,Km, such that V = ∪∞
m=1Km. Hint: Let
Cm ≡{
x ∈ Rn : dist(x,VC)≥ 1
m
}where
dist(x,S)≡ inf{|y−x| such that y ∈ S}.
Consider Km ≡Cm∩B(0,m).
4. Let B(X ;Rn) be the space of functions f, mapping X to Rn such that
sup{|f(x)| : x ∈ X}< ∞.
Show B(X ;Rn) is a complete normed linear space if
||f|| ≡ sup{|f(x)| : x ∈ X}.
5. Let α ∈ [0,1]. Define, for X a compact subset of Rp,
Cα (X ;Rn)≡ {f ∈C (X ;Rn) : ρα (f)+ ||f|| ≡ ||f||α< ∞}
where||f|| ≡ sup{|f(x)| : x ∈ X}
and
ρα (f)≡ sup{ |f(x)− f(y)||x−y|α
: x,y ∈ X , x ̸= y}.
Show that (Cα (X ;Rn) , ||·||α) is a complete normed linear space.
6. Let {fn}∞n=1 ⊆Cα (X ;Rn) where X is a compact subset of Rp and suppose
||fn||α ≤M
for all n. Show there exists a subsequence, nk, such that fnk converges in C (X ;Rn).The given sequence is called precompact when this happens. (This also shows theembedding of Cα (X ;Rn) into C (X ;Rn) is a compact embedding.) Note that it islikely the case that Cα (X ;Rn) is not separable although it embedds continuouslyinto a nice separable space. In fact, Cα ([0,T ] ;Rn) can be shown to not be separable.See Definition 9.3.1 and the discussion which follows it.
7. Use the general Stone Weierstrass approximation theorem to prove Theorem 9.1.7.