6.7.E: Problems on Inverse and Implicit Functions, Open and Closed Maps
( \newcommand{\kernel}{\mathrm{null}\,}\)
Discuss: In Definition 1, ¯G can equivalently be replaced by G=G→p(δ) (an open globe).
Prove that if the set D is open (closed) in (S,ρ), then the map f:S→T is open (closed, respectively) on D iff fD(f restricted to D) has this property as a map of D into f[D].
[Hint: Use Theorem 4 in Chapter 3, §12.]
Complete the missing details in the proofs of Theorems 1-4.
Verify footnotes 2 and 3.
Show that a map f:E′→E may fail to be one-to-one on all of E′ even if f satisfies Theorem 1 near every →p∈E′. Nonetheless, show that this cannot occur if E′=E=E1.
[Hints: For the first part, take E′=C,f(x+iy)=ex(cosy+isiny). For the second, use Theorem 1 in Chapter 5, §2.]
(i) For maps f:E1→E1, prove that the existence of a bijective df(p;⋅) is equivalent to f′(p)≠0.
(ii) Let
f(x)=x+x2sin1x,f(0)=0.
Show that f′(0)≠0, and f∈CD1 near any p≠0; yet f is not one-to-one near 0. What is wrong?
Show that a map f:En(Cn)→En(Cn),f∈CD1, may be bijective even if det[f′(→p)]=0 at some →p, but then f−1 cannot be differentiable at →q=f(→p).
[Hint: For the first clause, take f(x)=x3,p=0; for the second, note that if f−1 is differentiable at →q, then Note 2 in §4 implies that det [df(→p;⋅)]⋅det[df−1(→q;⋅)]=1≠0, since f∘f−1 is the identity map.]
Prove Corollary 2 for the general case of complete E′ and E.
[Outline: Given a closed X⊆¯G, take any convergent sequence {→yn}⊆f[X]. By Problem 8 in Chapter 4, §8, f−1(→yn)=→xn is a Cauchy sequence in X (why?). By the completeness of E′,(∃→x∈X)→xn→→x (Theorem 4 of Chapter 3, §16). Infer that lim→yn=f(→x)∈f[X], so f[X] is closed.]
Prove that "the composite of two open (closed) maps is open (closed)." State the theorem precisely. Prove it also for the uniform Lipschitz property.
Prove in detail that f:(S,ρ)→(T,ρ′) is open on D⊆S iff f maps the interior of D into that of f[D]; that is, f[D0]⊆(f[D])0.
Verify by examples that f may be:
(i) closed but not open;
(ii) open but not closed.
[Hints: (i) Consider f= constant. (ii) Define f:E2→E1 by f(x,y)=x and let
D={(x,y)∈E2|y=1x,x>0};
use Theorem 4(iii) in Chapter 3, §16 and continuity to show that D is closed in E2, but f[D]=(0,+∞) is not closed in E1. However, f is open on all of E2 by Problem 8. (Verify!)]
Continuing Problem 9(ii), define f:En→E1 (or Cn→C) by f(→x)= xk for a fixed k≤n (the "kth projection map"). Show that f is open, but not closed, on En(Cn).
(i) In Example (a), take (p,q)=(5,0) or (−5,0). Are the conditions of Theorem 4 satisfied? Do the conclusions hold?
(ii) Verify Example (b).
(i) Treating z as a function of x and y, given implicitly by
f(x,y,z)=z3+xz2−yz=0,f:E3→E1,
discuss the choices of P and Q that satisfy Theorem 4. Find ∂z∂x and ∂z∂y.
(ii) Do the same for f(x,y,z)=exyz−1=0.
Given f:En(Cn)→Em(Cm),n>m, prove that if f∈CD1 on a globe G,f cannot be one-to-one.
[Hint for f:E2→E1: If, say, D1f≠0 on G,setF(x,y)=(f(x,y),y).]
Suppose that f satisfies Theorem 1 for every →p in an open set A⊆E′, and is one-to-one on A (cf. Problem 4). Let g=f−1A (restrict f to A and take its inverse). Show that f and g are open and of class CD1 on A and f[A], respectively.
Given →v∈E and a scalar c≠0, define T→v:E→E ("translation by →v") and Mc:E→E ("dilation by c"), by setting
T→v(→x)=→x+→v and Mc(→x)=c→x.
Prove the following.
(i) T→v and T−1→v(=T−→v) are bijective, continuous, and "clopen" on E; so also are Mc and M−1c(=M1/c).
(ii) Similarly for the Lipschitz property on E.
(iii) If G=G→q(δ)⊂E, then T→v[G]=G→q+→v(δ), and Mc[G]=Gc→q(|cδ|).
(iv) If f:E′→E is linear, and →v=f(→p) for some →p∈E′, then T→v∘f=f∘T′→p and Mc∘f=f∘M′c, where T′→p and M′c are the corresponding maps on E′. If, further, f is continuous at →p, it is continuous on all of E′.
[Hint for (iv): Fix any →x∈E′. Set →v=f(→x−→p),g=T→v∘f∘T′→p−→x. Verify that g=f,T′→p−→x(→x)=→p, and g is continuous at →x.]
Show that if f:E′→E is linear and if f[G∗] is open in E for some G∗=G→p(δ)⊆E′, then
(i) f is open on all of E′;
(ii) f is onto E.
[Hints: (i) By Problem 8, it suffices to show that the set f[G] is open, for any globe G (why?). First take G=G→0(δ). Then use Problems 7 and 15(i)-(iv), with suitable →v and c.
(ii) To prove E=f[E′], fix any →y∈E. As \(f=G_{\overrightarrow{0} (\delta\) is open, it contains a globe G′=G→0(r). For small c,c→y∈G′⊆f[E′]. Hence →y∈f[E′] (Problem 10 in §2).]
Continuing Problem 16, show that if f is also one-to-one on G∗, then
f:E′⟷ onto E,
f∈L(E′,E),f−1∈L(E,E′),f is clopen on E′, and f−1 is so on E.
[Hints: To prove that f is one-to-one on E′, let f(→x)=→y for some →x,→x′∈E′. Show that
(∃c,ε>0)c→y∈G→0(ε)⊆f[G→0(δ)] and f(c→x+→p)=f(c→x′+→p)∈f[G→p(δ)]=f[G∗].
Deduce that c→x+→p=c→x′+→p and →x=→x′. Then use Problem 15(v) in Chapter 4, §2, and Note 1.]
A map
f:(S,ρ)⟷ onto (T,ρ′)
is said to be bicontinuous, or a homeomorphism, (from S onto T) iff both f and f−1 are continuous. Assuming this, prove the following.
(i) xn→p in S iff f(xn)→f(p) in T;
(ii) A is closed (open, compact, perfect) in S iff f[A] is so in T;
(iii) B=¯A in S iff f[B]=¯f[A] in T;
(iv) B=A0 in S iff f[B]=(f[A])0 in T;
(v) A is dense in B (i.e., A⊆B⊆¯A⊆S) in (S,ρ) iff f[A] is dense in f[B]⊆(T,ρ′).
[Hint: Use Theorem 1 of Chapter 4, §2, and Theorem 4 in Chapter 3, §16, for closed sets; see also Note 1.]
Given A,B⊆E,→v∈E and a scalar c, set
A+→v={→x+→v|→x∈A} and cA={c→x|→x∈A}.
Assuming c≠0, prove that
(i) A is closed (open, compact, perfect) in E iff cA+→v is;
(ii) B=¯A iff cB+→v=¯cA+→v;
(iii) B=A0 iff cB+→v=(cA+→v)0;
(iv) A is dense in B iff cA+→v is dense in cB+→v.
[Hint: Apply Problem 18 to the maps T→v and Mc of Problem 15, noting that A+→v=T→v[A] and cA=Mc[A].]
Prove Theorem 2, for a reduced δ, assuming that only one of E′ and Eis En(Cn), and the other is just complete.
[Hint: If, say, E=En(Cn), then f[¯G] is compact (being closed and bounded), and so is ¯G=f−1[f[¯G]]. (Why?) Thus the Lemma works out as before, i.e., f[G]⊇G¯q(α).
Now use the continuity of f to obtain a globe G′=G→p(δ′)⊆G such that f[G′]⊆G→q(α). Let g=f−1G, further restricted to G→q(α). Apply Problem 15(v) in Chapter 4, §2, to g, with S=G→q(α),T=E′.]