6.7.E: Problems on Inverse and Implicit Functions, Open and Closed Maps
Discuss: In Definition 1, \(\overline{G}\) can equivalently be replaced by \(G=G_{\vec{p}}(\delta)\) (an open globe).
Prove that if the set \(D\) is open (closed) in \((S, \rho),\) then the map \(f : S \rightarrow T\) is open (closed, respectively) on \(D\) iff \(f_{D}(f \text { restricted to } D)\) 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 \(f : E^{\prime} \rightarrow E\) may fail to be one-to-one on all of \(E^{\prime}\) even if \(f\) satisfies Theorem 1 near every \(\vec{p} \in E^{\prime}.\) Nonetheless, show that this cannot occur if \(E^{\prime}=E=E^{1}\).
[Hints: For the first part, take \(E^{\prime}=C, f(x+i y)=e^{x}(\cos y+i \sin y).\) For the second, use Theorem 1 in Chapter 5, §2.]
(i) For maps \(f : E^{1} \rightarrow E^{1},\) prove that the existence of a bijective \(d f(p ; \cdot)\) is equivalent to \(f^{\prime}(p) \neq 0\).
(ii) Let
\[f(x)=x+x^{2} \sin \frac{1}{x}, \quad f(0)=0.\]
Show that \(f^{\prime}(0) \neq 0,\) and \(f \in C D^{1}\) near any \(p \neq 0;\) yet \(f\) is not one-to-one near 0. What is wrong?
Show that a map \(f : E^{n}\left(C^{n}\right) \rightarrow E^{n}\left(C^{n}\right), f \in C D^{1},\) may be bijective even if \(\operatorname{det}\left[f^{\prime}(\vec{p})\right]=0\) at some \(\vec{p},\) but then \(f^{-1}\) cannot be differentiable at \(\vec{q}=f(\vec{p}).\)
[Hint: For the first clause, take \(f(x)=x^{3}, p=0;\) for the second, note that if \(f^{-1}\) is differentiable at \(\vec{q},\) then Note 2 in §4 implies that det \([d f(\vec{p} ; \cdot)] \cdot \operatorname{det}\left[d f^{-1}(\vec{q} ; \cdot)\right]=1 \neq 0,\) since \(f \circ f^{-1}\) is the identity map.]
Prove Corollary 2 for the general case of complete \(E^{\prime}\) and \(E\).
[Outline: Given a closed \(X \subseteq \overline{G},\) take any convergent sequence \(\left\{\vec{y}_{n}\right\} \subseteq f[X].\) By Problem 8 in Chapter 4, §8, \(f^{-1}\left(\vec{y}_{n}\right)=\vec{x}_{n}\) is a Cauchy sequence in \(X\) (why?). By the completeness of \(E^{\prime},(\exists \vec{x} \in X) \vec{x}_{n} \rightarrow \vec{x}\) (Theorem 4 of Chapter 3, §16). Infer that \(\lim \vec{y}_{n}=f(\vec{x}) \in f[X],\) 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 \(f :(S, \rho) \rightarrow\left(T, \rho^{\prime}\right)\) is open on \(D \subseteq S\) iff \(f\) maps the interior of \(D\) into that of \(f[D];\) that is, \(f\left[D^{0}\right] \subseteq(f[D])^{0}\).
Verify by examples that \(f\) may be:
(i) closed but not open;
(ii) open but not closed.
[Hints: (i) Consider \(f=\) constant. (ii) Define \(f : E^{2} \rightarrow E^{1}\) by \(f(x, y)=x\) and let
\[D=\left\{(x, y) \in E^{2} | y=\frac{1}{x}, x>0\right\};\]
use Theorem 4(iii) in Chapter 3, §16 and continuity to show that \(D\) is closed in \(E^{2},\) but \(f[D]=(0,+\infty)\) is not closed in \(E^{1}.\) However, \(f\) is open on all of \(E^{2}\) by Problem 8. (Verify!)]
Continuing Problem 9(ii), define \(f : E^{n} \rightarrow E^{1}\) (or \(C^{n} \rightarrow C)\) by \(f(\vec{x})=\) \(x_{k}\) for a fixed \(k \leq n\) (the "\(k\)th projection map"). Show that \(f\) is open, but not closed, on \(E^{n}\left(C^{n}\right)\).
(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
\[f(x, y, z)=z^{3}+x z^{2}-y z=0, \quad f : E^{3} \rightarrow E^{1},\]
discuss the choices of \(P\) and \(Q\) that satisfy Theorem 4. Find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\).
(ii) Do the same for \(f(x, y, z)=e^{x y z}-1=0\).
Given \(f : E^{n}\left(C^{n}\right) \rightarrow E^{m}\left(C^{m}\right), n>m,\) prove that if \(f \in C D^{1}\) on a globe \(G, f\) cannot be one-to-one.
[Hint for \(f : E^{2} \rightarrow E^{1}:\) If, say, \(D_{1} f \neq 0\) on \(G, \operatorname{set} F(x, y)=(f(x, y), y)\).]
Suppose that \(f\) satisfies Theorem 1 for every \(\vec{p}\) in an open set \(A \subseteq E^{\prime}\), and is one-to-one on \(A\) (cf. Problem 4). Let \(g=f_{A}^{-1}\) (restrict \(f\) to \(A\) and take its inverse). Show that \(f\) and \(g\) are open and of class \(C D^{1}\) on \(A\) and \(f[A],\) respectively.
Given \(\vec{v} \in E\) and a scalar \(c \neq 0,\) define \(T_{\vec{v}} : E \rightarrow E\) ("translation by \(\vec{v}\)") and \(M_{c} : E \rightarrow E\) ("dilation by \(c\)"), by setting
\[T_{\vec{v}}(\vec{x})=\vec{x}+\vec{v} \text { and } M_{c}(\vec{x})=c \vec{x}.\]
Prove the following.
(i) \(T_{\vec{v}}\) and \(T_{\vec{v}}^{-1}\left(=T_{-\vec{v}}\right)\) are bijective, continuous, and "clopen" on \(E;\) so also are \(M_{c}\) and \(M_{c}^{-1}\left(=M_{1 / c}\right).\)
(ii) Similarly for the Lipschitz property on \(E\).
(iii) If \(G=G_{\vec{q}}(\delta) \subset E,\) then \(T_{\vec{v}}[G]=G_{\vec{q}+\vec{v}}(\delta),\) and \(M_{c}[G]=G_{c \vec{q}}(|c \delta|)\).
(iv) If \(f : E^{\prime} \rightarrow E\) is linear, and \(\vec{v}=f(\vec{p})\) for some \(\vec{p} \in E^{\prime},\) then \(T_{\vec{v}} \circ f=f \circ T_{\vec{p}}^{\prime}\) and \(M_{c} \circ f=f \circ M_{c}^{\prime},\) where \(T_{\vec{p}}^{\prime}\) and \(M_{c}^{\prime}\) are the corresponding maps on \(E^{\prime}.\) If, further, \(f\) is continuous at \(\vec{p},\) it is continuous on all of \(E^{\prime}.\)
[Hint for (iv): Fix any \(\vec{x} \in E^{\prime}.\) Set \(\vec{v}=f(\vec{x}-\vec{p}), g=T_{\vec{v}} \circ f \circ T_{\vec{p}-\vec{x}}^{\prime}.\) Verify that \(g=f, T_{\vec{p}-\vec{x}}^{\prime}(\vec{x})=\vec{p},\) and \(g\) is continuous at \(\vec{x}\).]
Show that if \(f : E^{\prime} \rightarrow E\) is linear and if \(f\left[G^{*}\right]\) is open in \(E\) for some \(G^{*}=G_{\vec{p}}(\delta) \subseteq E^{\prime},\) then
(i) \(f\) is open on all of \(E^{\prime}\);
(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 \(G=G_{\overrightarrow{0}}(\delta).\) Then use Problems 7 and 15(i)-(iv), with suitable \(\vec{v}\) and \(c.\)
(ii) To prove \(E=f\left[E^{\prime}\right],\) fix any \(\vec{y} \in E.\) As \(f=G_{\overrightarrow{0} (\delta\) is open, it contains a globe \(G^{\prime}=G_{\overrightarrow{0}}(r).\) For small \(c, c \vec{y} \in G^{\prime} \subseteq f\left[E^{\prime}\right].\) Hence \(\vec{y} \in f\left[E^{\prime}\right]\) (Problem 10 in §2).]
Continuing Problem 16, show that if \(f\) is also one-to-one on \(G^{*},\) then
\[f : E^{\prime} \stackrel{\longleftrightarrow}{\text { onto }} E,\]
\(f \in L\left(E^{\prime}, E\right), f^{-1} \in L\left(E, E^{\prime}\right), f\) is clopen on \(E^{\prime},\) and \(f^{-1}\) is so on \(E\).
[Hints: To prove that \(f\) is one-to-one on \(E^{\prime},\) let \(f(\vec{x})=\vec{y}\) for some \(\vec{x}, \vec{x}^{\prime} \in E^{\prime}\). Show that
\[(\exists c, \varepsilon>0) \quad c \vec{y} \in G_{\overrightarrow{0}}(\varepsilon) \subseteq f\left[G_{\overrightarrow{0}}(\delta)\right] \text { and } f(c \vec{x}+\vec{p})=f\left(c \vec{x}^{\prime}+\vec{p}\right) \in f\left[G_{\vec{p}}(\delta)\right]=f\left[G^{*}\right].\]
Deduce that \(c \vec{x}+\vec{p}=c \vec{x}^{\prime}+\vec{p}\) and \(\vec{x}=\vec{x}^{\prime}.\) Then use Problem 15(v) in Chapter 4, §2, and Note 1.]
A map
\[f :(S, \rho)\stackrel{\longleftrightarrow}{\text { onto }} (T, \rho^{\prime})\]
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) \(x_{n} \rightarrow p\) in \(S\) iff \(f\left(x_{n}\right) \rightarrow f(p)\) in \(T\);
(ii) \(A\) is closed (open, compact, perfect) in \(S\) iff \(f[A]\) is so in \(T\);
(iii) \(B=\overline{A}\) in \(S\) iff \(f[B]=\overline{f[A]}\) in \(T\);
(iv) \(B=A^{0}\) in \(S\) iff \(f[B]=(f[A])^{0}\) in \(T\);
(v) \(A\) is dense in \(B\) (i.e., \(A \subseteq B \subseteq \overline{A} \subseteq S\)) in \((S, \rho)\) iff \(f[A]\) is dense in \(f[B] \subseteq\left(T, \rho^{\prime}\right).\)
[Hint: Use Theorem 1 of Chapter 4, §2, and Theorem 4 in Chapter 3, §16, for closed sets; see also Note 1.]
Given \(A, B \subseteq E, \vec{v} \in E\) and a scalar \(c,\) set
\[A+\vec{v}=\{\vec{x}+\vec{v} | \vec{x} \in A\} \text { and } c A=\{c \vec{x} | \vec{x} \in A\}.\]
Assuming \(c \neq 0,\) prove that
(i) \(A\) is closed (open, compact, perfect) in \(E\) iff \(c A+\vec{v}\) is;
(ii) \(B=\overline{A}\) iff \(c B+\vec{v}=\overline{c A+\vec{v}}\);
(iii) \(B=A^{0}\) iff \(c B+\vec{v}=(c A+\vec{v})^{0}\);
(iv) \(A\) is dense in \(B\) iff \(c A+\vec{v}\) is dense in \(c B+\vec{v}\).
[Hint: Apply Problem 18 to the maps \(T_{\vec{v}}\) and \(M_{c}\) of Problem 15, noting that \(A+\vec{v}= T_{\vec{v}}[A]\) and \(c A=M_{c}[A].\)]
Prove Theorem 2, for a reduced \(\delta,\) assuming that only one of \(E^{\prime}\) and \(E\)is \(E^{n}\left(C^{n}\right),\) and the other is just complete.
[Hint: If, say, \(E=E^{n}\left(C^{n}\right),\) then \(f[\overline{G}]\) is compact (being closed and bounded), and so is \(\overline{G}=f^{-1}[f[\overline{G}]].\) (Why?) Thus the Lemma works out as before, i.e., \(f[G] \supseteq G_{\overline{q}}(\alpha)\).
Now use the continuity of \(f\) to obtain a globe \(G^{\prime}=G_{\vec{p}}\left(\delta^{\prime}\right) \subseteq G\) such that \(f\left[G^{\prime}\right] \subseteq G_{\vec{q}}(\alpha).\) Let \(g=f_{G}^{-1},\) further restricted to \(G_{\vec{q}}(\alpha).\) Apply Problem 15(v) in Chapter 4, §2, to \(g,\) with \(S=G_{\vec{q}}(\alpha), T=E^{\prime}\).]