84 CHAPTER 3. STONE WEIERSTRASS APPROXIMATION THEOREM
•
•
•(⃗0,2)
(⃗0,1)•
p
θ(p)
Rn
This map θ is one to one onto Sp \{(⃗
0,2)}
. More precisely, if you have (⃗a,an+1)
on Sp \(⃗
0,2)
to get θ−1 (⃗a,an+1) , you form the line from
(⃗0,2)
through this point and
see where it hits Rp. The line is(⃗
0,2)+ t((⃗a,an+1)−
(⃗0,2))
and it hits Rp when 2+
t (an+1−2) = 0 which is when t = 22−an+1
. Thus θ−1 (⃗a,an+1) =
(2⃗a
2−an+1,0). From this
formula, it is clear that θ−1 is continuous and one to one. It is also onto because if x⃗ ∈ Rp
you can take the line from(⃗
0,2)
to (⃗x,0) and where it intersects Sp is the point which iswanted. It is also easy to see from this that θ is continuous. Indeed, suppose x⃗k → x⃗ inRp. Does it follow that θ (⃗xk)→ θ (⃗x)? We know that {⃗xk} is bounded since it converges.Therefore, there is an open ball, B
((⃗0,2),r)
such that θ (⃗xk) ∈ Sp \B((⃗
0,2),r)≡ K a
compact set. If θ (⃗xk) fails to converge to θ (⃗x) , then there is a subsequence, still denotedas θ (⃗xk) such that θ (⃗xk)→ y ∈ K where y ̸= θ (⃗x) . But then, the continuity of θ
−1 impliesxk→ θ
−1 (y) and so θ−1 (y) = x which implies y = θ (x) , a contradiction. Thus both θ and
θ−1 are continuous, one to one and onto mappings between Rp and Sp \
{(⃗0,2)}
.
Theorem 3.3.3 Let A be an algebra of functions of C0 (X ,R) which separates thepoints of the closed set X ⊆Rp and annihilates no point of X. Then A is dense in C0 (X ;R).
Proof: Ã denote all finite linear combinations of the form{n
∑i=1
ci f̃i + c0 : f ∈A , ci ∈ R
}
where for f ∈C0 (X ;R) ,
f̃ (x)≡
{f(θ−1 (x)
)if x ∈ θ (X)
0 if x =(⃗
0,2) .
Then à is obviously an algebra of functions in C (Sp;R). It separates points because thisis true of A . Similarly, it annihilates no point because of the inclusion of c0 an arbitraryelement of R in the definition of à above. Therefore from Theorem 3.2.4, à is dense inC (Sp;R) . Letting f ∈C0 (X ;R) , it follows f̃ ∈C (Sp;R) . It is clearly continuous on θ (X) .
What about at(⃗
0,2)
? If you have xn→(⃗
0,2), then
∣∣θ−1 (xn)∣∣→ ∞ and therefore, since
f ∈C0, f(θ−1 (xn)
)≡ f̃ (xn)→ 0≡ f̃
((⃗0,2))
and so indeed f̃ is in C (Sp;R) as claimed.
Thus there exists a sequence {hn} ⊆ Ã such that hn converges uniformly to f̃ . Now hn is