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Theorem 1.95 (Picard, Banach, Nadler [739]). Let \( \left( {X, d}\right) \) be a complete metric space and let \( F : X \rightrightarrows X \) be a multivalued contraction with nonempty closed values. Then \( F \) has a fixed point: there exists \( u \in X \) such that \( u \in F\left( u\right) \) . | Proof. Let us define \( f \) on \( X \) by \( f\left( x\right) = d\left( {x, F\left( x\right) }\right) \mathrel{\text{:=}} \inf \{ d\left( {x, y}\right) : y \in F\left( x\right) \} \) . Since\n\n\[ d\left( {x, F\left( x\right) }\right) \leq d\left( {x,{x}^{\prime }}\right) + d\left( {{x}^{\prime }, F\left( {x}^{\prime ... | Yes |
Theorem 1.96 (Caristi, Kirk). Let \( X \) be a complete metric space and let \( F \) : \( X \rightrightarrows X \) be a multimap with nonempty values. Suppose there exists some lower semicontinuous function \( h : X \rightarrow {\overline{\mathbb{R}}}_{ + } \mathrel{\text{:=}} {\mathbb{R}}_{ + } \cup \{ + \infty \} \) ... | Proof. Applying Theorem 1.87 with \( f \mathrel{\text{:=}} h,\gamma \mathrel{\text{:=}} 1 \), we get some \( u \in X \) such that \( h\left( u\right) < h\left( x\right) + d\left( {x, u}\right) \) for all \( x \neq u \) . If we could find \( v \in F\left( u\right) \) with \( v \neq u \), taking \( x \mathrel{\text{:=}} ... | Yes |
Theorem 1.98 (Drop theorem [253]). Let \( E \) be a nonempty complete subset of a normed space \( X \) and let \( B \) be a nonempty, bounded, closed, convex subset of \( X \) such that \( \delta \mathrel{\text{:=}} \operatorname{gap}\left( {B, E}\right) > 0 \) . Then there exists some \( e \in E \) such that \( D\left... | Proof. Let \( \beta \mathrel{\text{:=}} \operatorname{diam}\left( B\right) \mathrel{\text{:=}} \sup \{ d\left( {w, x}\right) : w, x \in B\} \) be the diameter of \( B \) and let \( \gamma > 0 \) be such that \( \gamma \left( {1 + \beta /\delta }\right) < 1 \) . Let us apply Theorem 1.88 to the function \( f \mathrel{\t... | Yes |
Lemma 1.99. Let \( E \) be a nonempty complete subset of a normed space \( X \), let \( \bar{w} \in \) \( X \smallsetminus E,\bar{e} \in E, s > r > 0 \) such that \( B\left\lbrack {\bar{w}, s}\right\rbrack \cap E = \varnothing \), and let \( B = B\left\lbrack {\bar{w}, r}\right\rbrack, C \mathrel{\text{:=}} {\mathbb{R}... | Proof. Theorem 1.98 yields some \( e \in E \cap D\left( {\bar{e}, B}\right) \) such that \( D\left( {e, B}\right) \cap E = \{ e\} \) . Let \( b \in B, t \in (0,1\rbrack \) be such that \( e = \left( {1 - t}\right) b + t\bar{e} \) . Then since \( {tB} + \left( {1 - t}\right) b \subset B \), we have\n\n\[ B - \bar{e} = B... | Yes |
Corollary 1.100. For every nonempty complete subset \( E \) of a normed space \( X \), the set \( S \) of support points of \( E \) is dense in the boundary \( \operatorname{bdry}\left( E\right) \) of \( E \) . | Proof. The conclusion is obvious if the boundary of \( E \) is empty. Let \( \bar{e} \) be a boundary point of \( E \) and let \( \alpha > 0 \) be given. There exists some \( \bar{w} \in B\left( {\bar{e},\alpha /2}\right) \smallsetminus E \) . Let \( s > 0 \) be the radius of a closed ball with center \( \bar{w} \) con... | Yes |
Theorem 1.102 (Bishop-Phelps). For every bounded, closed, convex subset \( C \) of a Banach space \( X \), the set of continuous linear forms that attain their maximum on \( C \) is dense in \( {X}^{ * } \) . | Proof. Let \( h \in {X}^{ * } \) and let \( \varepsilon > 0 \) be given. Let \( g \) be the indicator function of \( C \) and let \( f \mathrel{\text{:=}} g - h \) . Let \( \bar{x} \in C \) be such that \( f\left( \bar{x}\right) < \inf f\left( C\right) + \alpha \) with \( \alpha \mathrel{\text{:=}} 1 \) . Taking \( r >... | Yes |
Corollary 1.107 (Closed graph theorem). Every linear map with closed graph between two Banach spaces is continuous. | Proof. Let \( B : Y \rightarrow Z \) be such a map. The graph \( X \) of \( B \), being a closed linear subspace of \( Y \times Z \), is a Banach space, and \( A : \left( {y,{By}}\right) \mapsto y \) is a continuous bijection from \( X \) onto \( Y \) . Its inverse \( y \mapsto \left( {y,{By}}\right) \) being continuou... | Yes |
Lemma 1.108. Let \( X, Y \) be Banach spaces, let \( A : X \rightarrow Y \) be a surjective continuous linear map, and let \( \ell \in {X}^{ * } \) be such that \( \ell \left( x\right) = 0 \) for all \( x \in N \mathrel{\text{:=}} \ker A \) . Then there exists some \( {y}^{ * } \) in the dual \( {Y}^{ * } \) of \( Y \)... | Proof. Since \( A \) is surjective and since for every \( x,{x}^{\prime } \in X \) satisfying \( A\left( x\right) = A\left( {x}^{\prime }\right) \) one has \( \ell \left( x\right) = \ell \left( {x}^{\prime }\right) \), there exists a map \( k : Y \rightarrow \mathbb{R} \) such that \( \ell = k \circ A \) . It is easy t... | Yes |
Proposition 1.110 (Error bound property). Let \( f : X \rightarrow {\overline{\mathbb{R}}}_{ + } \) be a nonnegative lower semicontinuous function on a complete metric space \( \left( {X, d}\right) \) and let \( S \mathrel{\text{:=}} \) \( {f}^{-1}\left( {\{ 0\} }\right) \) . Let \( x \in \operatorname{dom}f, c > 0 \),... | Proof. In Theorem 1.88 take \( \bar{x} \mathrel{\text{:=}} x,\gamma \in \left( {0, c}\right) \) with \( f\left( x\right) /\gamma < \rho \) . Let \( u \in X \) be given by the conclusion of Theorem 1.88, i.e., such that\n\n\[ f\left( u\right) + {\gamma d}\left( {u, x}\right) \leq f\left( x\right) ,\;f\left( u\right) < f... | Yes |
Proposition 1.111 (Steepness principle [225]). Let \( \left( {X, d}\right) \) be a complete metric space and let \( f : X \rightarrow {\overline{\mathbb{R}}}_{ + }, S \mathrel{\text{:=}} {f}^{-1}\left( {\{ 0\} }\right) \) . Suppose that \( {f}^{-1} \) is closed at 0 and that for some \( s \in \left( {0,1}\right) \) the... | Proof. We may assume that \( f\left( x\right) > 0 \) and \( B\left( {x, f\left( x\right) /s}\right) \cap S = \varnothing \) since otherwise the inequality is trivial. Let us show that \( B\left\lbrack {x, f\left( x\right) /s}\right\rbrack \cap S \neq \varnothing \) . Starting with \( {u}_{0} \mathrel{\text{:=}} x \) , ... | Yes |
Theorem 1.113 (Decrease principle). Let \( X \) be a complete metric space, let \( f \) : \( X \rightarrow {\overline{\mathbb{R}}}_{ + } \) be a function such that \( {f}^{-1} \) is closed at 0, and let \( S \mathrel{\text{:=}} {f}^{-1}\left( {\{ 0\} }\right) \) . Let \( {\delta }_{f} : X \rightarrow {\overline{\mathbb... | Proof. If for all \( s \in \left( {0, c}\right) \) one has \( B\left( {x, f\left( x\right) /s}\right) \cap S \neq \varnothing \), then the result holds. Thus, we consider the case that there exists some \( \bar{s} \in \left( {0, c}\right) \) such that \( B\left( {x, f\left( x\right) /\bar{s}}\right) \cap S = \varnothin... | Yes |
Theorem 1.114 (Local decrease principle). Let \( X \) be a complete metric space, let \( f : X \rightarrow {\overline{\mathbb{R}}}_{ + } \) be such that \( {f}^{-1} \) is closed at 0 on \( X \), and let \( \bar{x} \in S \mathrel{\text{:=}} {f}^{-1}\left( {\{ 0\} }\right), c > 0 \) , and \( \rho \in (0, + \infty \rbrack... | Proof. Given \( x \in B\left( {\bar{x},\rho }\right) \) satisfying \( f\left( x\right) \geq {c\rho } \), we obviously have \( d\left( {x, S}\right) \leq \) \( {c}^{-1}f\left( x\right) \), since \( d\left( {x, S}\right) \leq d\left( {x,\bar{x}}\right) < \rho \) . Now, for \( x \in B\left( {\bar{x},\rho }\right) \) satis... | Yes |
Lemma 1.117. Let \( X \) be a complete metric space and let \( f \) be a lower semicontinuous function on the open ball \( B \mathrel{\text{:=}} B\left( {x, r}\right) \) . Suppose \( \inf f\left( B\right) > - \infty \) and let \( \beta \mathrel{\text{:=}} f\left( x\right) - \inf f\left( B\right) \) . Then for all \( t ... | Proof. We may suppose \( \beta < + \infty \) . We apply Theorem 1.88 to the restriction of \( f \) to the closed ball \( B\left\lbrack {x,{rs}}\right\rbrack \), where \( s \in \left( {t,1}\right) \) . Then there exists \( u \in B\left\lbrack {x,{rt}}\right\rbrack \) satisfying \( f\left( u\right) \leq f\left( x\right) ... | Yes |
Theorem 1.118 (Parameterized decrease principle). Let \( W \) be a topological space and let \( X \) be a complete metric space. Let \( f : W \times X \rightarrow {\overline{\mathbb{R}}}_{ + } \) and let \( \left( {\bar{w},\bar{x}}\right) \in \) \( S \mathrel{\text{:=}} \left\{ {\left( {w, x}\right) \in W \times X : f\... | Proof. Let \( U, c, r \) be as in the assumptions. Assumption (b) means that \( {q}_{w} \mathrel{\text{:=}} \) \( 2\left( {c + 1}\right) d\left( {\left( {\bar{x},0}\right) ,\operatorname{epi}{f}_{w}}\right) \) has limit 0 as \( w \rightarrow 0 \) . Assumption (c) implies that \( \bar{x} \in S\left( w\right) \) for all ... | Yes |
Proposition 1.120. Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be a bounded-below lower semicontinuous function on a complete metric space \( X \) and let \( \ell \mathrel{\text{:=}} \inf f\left( X\right) \in \mathbb{R} \) . If \( f \) satisfies the Palais-Smale condition \( \left( {\mathrm{{PS}}}_{\ell }\right)... | Proof. Let \( \left( {w}_{n}\right) \) be a minimizing sequence of \( f \) : setting \( {\varepsilon }_{n} \mathrel{\text{:=}} f\left( {w}_{n}\right) - \ell \) one has \( \left( {\varepsilon }_{n}\right) \rightarrow 0 \) . Taking \( r = 2, t = 1/2, x = {w}_{n} \) in Lemma 1.117, we can find \( {u}_{n} \in X \) such tha... | Yes |
Proposition 1.121 (Penalization lemma). Let \( A \) be a nonempty subset of a metric space \( X \) and let \( f : X \rightarrow \mathbb{R} \) be a Lipschitzian function with rate \( r \) . Then for every \( s \geq r \)\n\n\[ \mathop{\inf }\limits_{{x \in A}}f\left( x\right) = \mathop{\inf }\limits_{{x \in X}}\left( {f\... | Proof. Since \( {f}_{s} = f \) on \( A \), we have \( m \mathrel{\text{:=}} \inf f\left( A\right) \geq \inf {f}_{s}\left( X\right) \) . If we had strict inequality we could find \( x \in X \) such that \( {f}_{s}\left( x\right) < m \) . Then we would have \( s{d}_{A}\left( x\right) < \) \( m - f\left( x\right) \), so t... | Yes |
Lemma 1.123. (a) If \( h \) is uniformly lower semicontinuous around \( Z \), then \( { \land }_{Z}h = \) \( \inf h\left( Z\right) \) . | Proof. (a) The relation \( { \land }_{Z}h = \inf h\left( Z\right) \) is obvious if \( { \land }_{Z}h = + \infty \) . Thus, it suffices to prove that for every \( s > r > { \land }_{Z}h \) one has \( s > \inf h\left( Z\right) \) . Let \( \varepsilon \in \left( {0, s - r}\right) \) with \( 1/\varepsilon > r \) and let \(... | Yes |
Lemma 1.124. Let \( \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) be a (lower) coherent family of functions on \( X \) in the sense that for every sequences \( {\left( {x}_{1, n}\right) }_{n},\ldots ,{\left( {x}_{k, n}\right) }_{n} \) satisfying \( {\left( d\left( {x}_{i, n},{x}_{j, n}\right) \right) }_{n} \rightarrow 0 \... | Proof. Let \( f \mathrel{\text{:=}} {f}_{1} + \cdots + {f}_{k} \) and let \( h : {X}^{k} \rightarrow {\mathbb{R}}_{\infty } \) be defined for \( y \mathrel{\text{:=}} \left( {{y}_{1},\ldots ,{y}_{k}}\right) \) by \( h\left( y\right) \mathrel{\text{:=}} {f}_{1}\left( {y}_{1}\right) + \cdots + {f}_{k}\left( {y}_{k}\right... | Yes |
Lemma 1.125. Let \( \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) be a coherent family of functions on \( X \) and let \( {f}_{k + 1} \) be uniformly continuous. Then \( \left( {{f}_{1},\ldots ,{f}_{k + 1}}\right) \) is a coherent family. | Proof. Given sequences \( \left( {x}_{1, n}\right) ,\ldots ,\left( {x}_{k + 1, n}\right) \) satisfying \( \left( {d\left( {{x}_{i, n},{x}_{j, n}}\right) }\right) \rightarrow 0 \) for \( i, j \in \) \( {\mathbb{N}}_{k + 1} \), picking \( \left( {\varepsilon }_{n}\right) \rightarrow 0,\left( {x}_{n}\right) \) such that \... | Yes |
Proposition 1.129. (a) Every family \( \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) of lower semicontinuous functions on \( X \) all but one of which are locally Lipschitzian around \( \bar{x} \) is linearly coherent. | Proof. It suffices to prove that if \( f \) is lower semicontinuous and if \( g \) is Lipschitzian around \( \bar{x} \), then \( \left( {f, g}\right) \) is linearly coherent around \( \bar{x} \) . Then assertions (a) and (b) follow by induction on \( k \) . Since (1.36) is preserved when one changes the norm in \( X \t... | Yes |
Proposition 1.130. One always has \( m \leq { \land }_{g}h \) . If \( m > - \infty \), equality holds. | Proof. Let us first prove that for all \( c > 0 \) we have \( {m}_{c} \leq { \land }_{g}h \) . We may suppose that \( { \land }_{g}h < + \infty \) . Let \( s > r > { \land }_{g}h \) . By definition of \( { \land }_{g}h \), for all \( \delta > 0 \), there exist some \( x \in X, y \in Y \) satisfying \( d\left( {g\left( ... | Yes |
Proposition 1.132. Let \( f, g \) be bounded below, or more generally, let them be such that \( {m}_{b} \mathrel{\text{:=}} \inf {p}_{b}\left( {X \times X}\right) > - \infty \) for some \( b > 0 \) . Suppose \( \land \left( {f, g}\right) < \infty \) . Then given a sequence \( \left( {\varepsilon }_{n}\right) \rightarro... | Proof. Since \( {k}_{X} \) is a forcing function, the result stems from the inequalities\n\n\[ \land \left( {f, g}\right) + {\varepsilon }_{n} \geq \inf {p}_{n} + {\varepsilon }_{n} \geq f\left( {x}_{n}\right) + g\left( {y}_{n}\right) + n{k}_{X}\left( {{x}_{n},{y}_{n}}\right) \geq {m}_{b} + \left( {n - b}\right) {k}_{X... | Yes |
Theorem 1.133. Let \( f, h, g,{p}_{t} \) be as above and let \( \bar{x} \) be a robust minimizer of \( f + \) \( h \circ g \) with \( \left( {f + h \circ g}\right) \left( \bar{x}\right) \) finite. Suppose \( g \) is continuous at \( \bar{x} \) and, for some \( b > 0 \) , \( {m}_{b} \mathrel{\text{:=}} \inf {p}_{b}\left... | Proof. Let \( m \mathrel{\text{:=}} f\left( \bar{x}\right) + h\left( \bar{y}\right) \) . Since \( \left( {{x}_{n},{y}_{n}}\right) \) is an \( {\varepsilon }_{n} \) -minimizer of \( {p}_{n} \), we have\n\n\[ m + {\varepsilon }_{n} \geq {m}_{n} + {\varepsilon }_{n} \geq {p}_{n}\left( {{x}_{n},{y}_{n}}\right) \geq {m}_{b}... | Yes |
Proposition 1.137. A multimap \( F : X \rightrightarrows Y \) is \( c \) -regular around \( \bar{z} \mathrel{\text{:=}} \left( {\bar{x},\bar{y}}\right) \in F \) if and only if there exists some \( \varepsilon > 0 \) such that \( F \) is metrically regular on the set\n\n\[ P \mathrel{\text{:=}} {P}_{\varepsilon } \mathr... | Proof. Regularity on \( {P}_{\varepsilon } \) being obviously necessary, let us prove that it is sufficient. Suppose \( F \) is metrically regular on the set \( {P}_{\varepsilon } \) for some \( \varepsilon > 0 \) . Take \( \delta > 0 \) such that \( \left( {c + 1}\right) \delta < {c\varepsilon } \) and let \( \left( {... | Yes |
Theorem 1.139. For a multimap \( F : X \rightrightarrows Y \) between two metric spaces, a subset \( P \) of \( X \times Y \), and a positive number \( c \), the following assertions are equivalent:\n\n(a) \( F \) is open at the linear rate \( a \mathrel{\text{:=}} {c}^{-1} \) on \( P \) ;\n\n(b) \( {F}^{-1} : Y \right... | Proof. The equivalence (b) \( \Leftrightarrow \) (c) consists in taking \( Z \mathrel{\text{:=}} X, M \mathrel{\text{:=}} {F}^{-1}, z = x \) to pass from (1.40) to (1.43) and vice versa.\n\n(c) \( \Rightarrow \) (a) Suppose \( F \) is \( c \) -metrically regular on \( P \) . Given \( r > 0,\left( {x, y}\right) \in F,\l... | Yes |
Proposition 1.143. Let \( X \) be a metric space, let \( Y \) be a normed space, let \( F : X \rightrightarrows Y \) be a multimap with convex values, and let \( \bar{x} \in X,\bar{y} \in F\left( \bar{x}\right) \). If \( F \) is pseudo-Lipschitzian around \( \left( {\bar{x},\bar{y}}\right) \), then for some ball \( B \... | Proof. Without loss of generality we may assume that \( \bar{y} = 0 \) . Taking \( x \mathrel{\text{:=}} \bar{x} \), and observing that \( \bar{y} \in F\left( \bar{x}\right) \cap B\left\lbrack {\bar{y}, r}\right\rbrack \), we get that for all \( {x}^{\prime } \in B\left\lbrack {\bar{x}, q}\right\rbrack, F\left( {x}^{\p... | Yes |
Proposition 1.144. Let \( F : X \rightrightarrows Y \) be a multimap with closed values between two metric spaces and let \( \bar{x} \in X,\bar{y} \in F\left( \bar{x}\right) \). Then \( F \) is subregular at \( \left( {\bar{x},\bar{y}}\right) \) (and \( {F}^{-1} \) is calm at \( \left( {\bar{y},\bar{x}}\right) ) \) if ... | Note that since \( {f}^{-1}\left( {\{ 0\} }\right) = {F}^{-1}\left( \bar{y}\right) \), the assumption made in Theorem 1.114 that \( {f}^{-1} \) is closed at 0 means that \( {F}^{-1} \) is closed at \( \bar{y} \), i.e., that \( \operatorname{cl}\left( {\operatorname{gph}{F}^{-1}}\right) \cap (\{ \bar{y}\} \times \) \( X... | No |
Corollary 1.145. Let \( F : X \rightrightarrows Y \) be a multimap with closed values between two metric spaces and let \( \bar{x} \in X,\bar{y} \in F\left( \bar{x}\right) \) . Suppose \( {F}^{-1} \) is closed at \( \bar{y} \) and there exist \( b > 0 \), a neighborhood \( U \) of \( \bar{x} \), and a decrease index \(... | Then \( F \) is subregular at \( \left( {\bar{x},\bar{y}}\right) \) (and \( {F}^{-1} \) is calm at \( \left( {\bar{y},\bar{x}}\right) \) ) and \[ \forall x \in U\;d\left( {x,{F}^{-1}\left( \bar{y}\right) }\right) \leq \frac{1}{b}d\left( {\bar{y}, F\left( x\right) }\right) . \] | Yes |
Theorem 1.146. Let \( F : X \rightrightarrows Y \) be a multimap with closed graph between two complete metric spaces and let \( b \) be a positive number. Endow \( X \times Y \) with the metric \( {d}_{c} \) with \( c \mathrel{\text{:=}} 1/b \) . Let \( f : Y \times X \rightarrow {\mathbb{R}}_{\infty } \) be given by ... | Proof. Let \( S \mathrel{\text{:=}} {f}^{-1}\left( {\{ 0\} }\right) \) and for \( w \in W, S\left( w\right) \mathrel{\text{:=}} \{ x \in X : \left( {w, x}\right) \in S\} \) . Since for \( x \in X, F\left( x\right) \) is closed, we have \( x \in S\left( w\right) \) if and only if \( w \in F\left( x\right) \), so that \(... | Yes |
Lemma 1.147. A function \( f : X \rightarrow {\mathbb{R}}_{\infty } \) that is bounded below is well-posed if and only if \( \operatorname{diam}\left( {{S}_{f}\left( \varepsilon \right) }\right) \rightarrow 0 \) as \( \varepsilon \rightarrow {0}_{ + } \) . | Proof. The condition is sufficient, as it implies that any minimizing sequence is a Cauchy sequence. It is also necessary: if there were \( \delta > 0 \) and a sequence \( \left( {\varepsilon }_{n}\right) \rightarrow {0}_{ + } \) such that \( \operatorname{diam}\left( {{S}_{f}\left( {\varepsilon }_{n}\right) }\right) >... | Yes |
Theorem 1.148. Let \( W \) be a topological space, let \( X \) be a complete metric space and let \( F : W \times X \rightarrow {\mathbb{R}}_{\infty } \) be such that for all \( w \in W \) the function \( {F}_{w} \) is bounded below on \( X \) and has a nonempty domain. If the following two conditions are satisfied, th... | Proof. Given a sequence \( \left( {r}_{n}\right) \) in \( \mathbb{P} \mathrel{\text{:=}} \left( {0, + \infty }\right) \) with limit 0, we set\n\n\[ G = \mathop{\bigcap }\limits_{{n \in \mathbb{N}}}W\left( {r}_{n}\right) \]\n\nso that by our assumptions, \( G \) is a generic subset of \( W \) . Let us show that for ever... | Yes |
Lemma 1.149. Let \( W \) be a topological space, let \( X \) be a metric space and let \( F \) : \( W \times X \rightarrow {\mathbb{R}}_{\infty } \) be such that for all \( w \in W \) the function \( {F}_{w} \) is bounded below on \( X \) and has a nonempty domain. If the mapping \( w \mapsto {F}_{w} \) is continuous f... | Proof. Let \( r > 0 \) and let \( w \in W\left( r\right) \) . There exist \( \varepsilon > 0 \) and \( a \in X \) such that \( S\left( {{F}_{w},\varepsilon }\right) \subset \) \( B\left( {a, r}\right) \) and there exists a neighborhood \( V \) of \( w \) such that \( {F}_{v} \in U\left( {{F}_{w},\varepsilon /3}\right) ... | Yes |
Lemma 1.150. The density assumption (b) of Theorem 1.148 is satisfied whenever the following condition holds: for all \( w \in W, V \in \mathcal{N}\left( w\right), r > 0 \) there exist \( \varepsilon > \eta > 0, a \in S\left( {{F}_{w},\eta }\right), v \in V \) such that\n\n\[ \n{F}_{v}\left( a\right) \leq {F}_{w}\left(... | Proof. In order to prove that for every \( r > 0 \) the set \( W\left( r\right) \) is dense in \( W \), let us show that for all \( w \in W, V \in \mathcal{N}\left( w\right) \), the set \( W\left( r\right) \cap V \) is nonempty. Taking \( \varepsilon > \eta > 0 \) , \( v \in V, a \in S\left( {{F}_{w},\eta }\right) \) a... | Yes |
Corollary 1.151 (Metric variational principle). Let \( \left( {W,\parallel \cdot \parallel }\right) \) be a bumpable space of bounded functions on \( X \) . Then given a bounded-below, proper, lower semicontinuous function \( f : X \rightarrow {\mathbb{R}}_{\infty } \) the set of \( g \in W \) such that \( f + g \) is ... | Proof. Again, let \( F : W \times X \rightarrow {\mathbb{R}}_{\infty } \) be given by \( F\left( {w, x}\right) = f\left( x\right) + w\left( x\right) \) . It suffices to check that assumption (1.53) implies the conditions of Lemma 1.150. Let \( w \in W \) , \( V \in \mathcal{N}\left( w\right), r > 0 \) be given. We pick... | Yes |
Theorem 1.152 (Deville-Godefroy-Zizler variational principle). Let \( X \) be a normed space and let \( W \) be a linear subspace of \( \mathrm{{BC}}\left( X\right) \) endowed with a norm \( \parallel \cdot \parallel \) stronger than the norm \( \parallel \cdot {\parallel }_{\infty } \) of uniform convergence, for whic... | Proof. It suffices to check that the family of functions\n\n\[ B = \{ s\bar{b}\left( {a + t \cdot }\right) : s \in \mathbb{R}, t > 0, a \in X\} \]\n\nmakes \( \left( {W,\parallel \cdot \parallel }\right) \) bumpable. Let \( \sigma > 0 \) be such that the support of \( \bar{b} \) is contained in \( B\left( {0,\sigma }\r... | Yes |
Theorem 1.153 (Stegall). Let \( Y \) be a nonempty closed and bounded subset of a Banach space \( X \) with the RNP and let \( f \) be a bounded-below lower semicontinuous function on \( Y \) . Then the set of \( {x}^{ * } \in {X}^{ * } \) such that \( f + {x}^{ * } \) is well-posed on \( Y \) is a generic subset of \(... | Proof. Let \( W = {X}^{ * } \) and let \( F : W \times Y \rightarrow {\mathbb{R}}_{\infty } \) be given by \( F\left( {w, y}\right) = f\left( y\right) + \langle w, y\rangle \) . Let us endow \( W \) with the dual norm. Then \( F \) is lower semicontinuous and \( w \mapsto {F}_{w} \) is continuous for the topology of un... | No |
Corollary 1.154 (Fabian). Let \( X \) be a Banach space with the RNP and let \( f \) be a lower semicontinuous, bounded-below, super-coercive function on \( X \) . Then the set of \( {x}^{ * } \in {X}^{ * } \) such that \( f + {x}^{ * } \) is well-posed on \( X \) is a generic subset of \( {X}^{ * } \) . | Proof. Let \( a > 0 \) with \( \mathop{\liminf }\limits_{{\parallel x\parallel \rightarrow \infty }}f\left( x\right) /\parallel x\parallel > a \), so that there exists \( r > 0 \) such that \( f\left( x\right) \geq a\parallel x\parallel \) for \( x \in X \smallsetminus r{B}_{X} \) . Since \( f \) is bounded below, addi... | Yes |
Theorem 1.155. Let \( X \) be a Banach space and let \( C \) be a bounded, closed, convex subset of \( X \) with the RNP. Then the set \( G \) of continuous linear forms on \( X \) that firmly expose \( C \) is a dense \( {\mathcal{G}}_{\delta } \) subset of \( {X}^{ * } \) . Moreover, \( C \) is the closed convex hull... | Proof. Let \( W \mathrel{\text{:=}} {X}^{ * } \), let \( F : W \times X \rightarrow \mathbb{R} \) be the evaluation, and for \( r > 0 \), let\n\n\[ \nW\left( r\right) \mathrel{\text{:=}} \left\{ {{x}^{ * } \in {X}^{ * } : \exists \varepsilon > 0,\exists a \in C, S\left( {{x}^{ * }, C,\varepsilon }\right) \subset B\left... | No |
Proposition 2.4. If \( T, U \) are open intervals of \( \mathbb{R} \), if \( g : T \rightarrow U \) is differentiable at \( \bar{t} \in T \), and if \( h : U \rightarrow X \) is differentiable at \( \bar{u} \mathrel{\text{:=}} g\left( \bar{t}\right) \), then \( f \mathrel{\text{:=}} h \circ g \) is differentiable at \(... | Proof. Let \( v \mathrel{\text{:=}} {h}^{\prime }\left( \bar{u}\right) \) and let \( \alpha : T \rightarrow \mathbb{R},\beta : U \rightarrow X \) be such that \( \alpha \left( t\right) \rightarrow 0 \) as \( t \rightarrow \bar{t},\beta \left( u\right) \rightarrow 0 \) as \( u \rightarrow \bar{u} \) with \( g\left( t\ri... | Yes |
Proposition 2.5 (Leibniz rule). Let \( X, Y, Z \) be normed spaces and let \( b : X \times Y \rightarrow Z \) be a continuous bilinear map. Given functions \( f : T \rightarrow X, g : T \rightarrow Y \) that are differentiable at \( t \), the function \( h : r \mapsto b\left( {f\left( r\right), g\left( r\right) }\right... | Proof. By assumption, there exist some \( \alpha : \left( {T - t}\right) \rightarrow X,\beta : \left( {T - t}\right) \rightarrow Y \) satisfying \( \alpha \left( s\right) \rightarrow 0,\beta \left( s\right) \rightarrow 0 \) as \( s \rightarrow 0 \) such that\n\n\[ \nf\left( {t + s}\right) = f\left( t\right) + s{f}^{\pr... | Yes |
Lemma 2.6. Let \( f : T \rightarrow \mathbb{R} \) be a continuous function on some interval \( T \mathrel{\text{:=}} \left\lbrack {a, b}\right\rbrack \) of \( \mathbb{R} \), with \( a < b \) . If \( f \) is differentiable on \( \left( {a, b}\right) \) then there exists some \( c \in \left( {a, b}\right) \) such that | \[ f\left( b\right) - f\left( a\right) = {f}^{\prime }\left( c\right) \left( {b - a}\right) . \] | Yes |
Let \( f : T \rightarrow X \) be continuous on \( T \mathrel{\text{:=}} \left\lbrack {a, b}\right\rbrack \), let \( m \in {\mathbb{R}}_{ + } \), and let \( D \) be a countable subset of \( T \). Suppose that for all \( t \in \left( {a, b}\right) \smallsetminus D \), \( f \) has a right derivative at \( t \) such that \... | \[ \parallel f\left( b\right) - f\left( a\right) \parallel \leq m\left( {b - a}\right) . \] | Yes |
Corollary 2.11. Let \( f : T \rightarrow X \) be continuous on \( T \mathrel{\text{:=}} \left\lbrack {a, b}\right\rbrack \), let \( v \in X, r \in {\mathbb{R}}_{ + } \), and let \( D \) be a countable subset of \( T \) . Suppose \( f \) has a right derivative on \( \left( {a, b}\right) \smallsetminus D \) such that \( ... | Proof. Define \( h : T \rightarrow X \) by \( h\left( t\right) \mathrel{\text{:=}} f\left( t\right) - {tv} \) . Then \( h \) is continuous and for \( t \in \) \( \left( {a, b}\right) \smallsetminus D \) one has \( \begin{Vmatrix}{{h}_{ + }^{\prime }\left( t\right) }\end{Vmatrix} = \begin{Vmatrix}{{f}_{ + }^{\prime }\le... | Yes |
Proposition 2.13. If \( {g}_{1} \) and \( {g}_{2} \) are two primitives of an arbitrary function \( f : T \rightarrow \) \( X \), then \( {g}_{1} - {g}_{2} \) is constant. | Proof. If \( {g}_{1} \) and \( {g}_{2} \) are two primitives of \( f \), then there exist countable subsets \( {D}_{1} \) and \( {D}_{2} \) of \( T \) such that \( {g}_{i} \) is differentiable on \( T \smallsetminus {D}_{i} \) and \( {g}_{i}^{\prime }\left( t\right) = f\left( t\right) \) for all \( t \in T \smallsetmin... | Yes |
Proposition 2.15. Let \( X \) be a Banach space and let \( T \) be a compact interval of \( \mathbb{R} \) . A function \( f : T \rightarrow X \) is regulated (resp. normalized regulated) if and only if it is the uniform limit of a sequence \( \left( {f}_{n}\right) \) of step functions (resp. normalized step functions). | It follows that a regulated function on \( T \) is bounded. | No |
Proposition 2.18. Given Banach spaces \( X, Y \) and \( A \in L\left( {X, Y}\right) \), for every \( f \in \) \( R\left( {T, X}\right) \) one has \( A \circ f \in R\left( {T, Y}\right) \) and \( {\int }_{T}A \circ f = A\left( {{\int }_{T}f}\right) \) . | Proof. The first assertion is a direct consequence of the definition. It can also be checked by taking a sequence \( \left( {f}_{n}\right) \) in \( S\left( {T, X}\right) \) that converges uniformly to \( f \) . Since the relation \( {\int }_{T}A \circ {f}_{n} = A\left( {{\int }_{T}{f}_{n}}\right) \) is immediate, the s... | Yes |
Theorem 2.19. For \( f \in R\left( {T, X}\right) \), the map \( g : t \mapsto {\int }_{a}^{t}f\left( s\right) \mathrm{d}s \) is a primitive of \( f \) on \( T \) . | Proof. Given \( t \in \lbrack a, b),\varepsilon > 0 \), let \( \delta \in \left( {0, b - t}\right) \) be such that \( \begin{Vmatrix}{f\left( {t + r}\right) - f\left( {t}_{ + }\right) }\end{Vmatrix} \leq \) \( \varepsilon \) for every \( r \in (0,\delta \rbrack \) . Since for \( c \mathrel{\text{:=}} f\left( {t}_{ + }\... | Yes |
Proposition 2.22 (Integration by parts). Let \( X, Y \), and \( Z \) be Banach spaces, let \( \left( {x, y}\right) \mapsto x * y \) be a continuous bilinear map from \( X \times Y \) into \( Z \), and let \( f : T \rightarrow X \) , \( g : T \rightarrow Y \) be primitives of regulated functions, with \( T \mathrel{\tex... | Proof. The functions \( t \mapsto f\left( t\right) * {g}^{\prime }\left( t\right) \) and \( t \mapsto {f}^{\prime }\left( t\right) * g\left( t\right) \) clearly have one-sided limits at all points of \( T \mathrel{\text{:=}} \left\lbrack {a, b}\right\rbrack \) . Moreover, their sum is the derivative of \( h : t \mapsto... | Yes |
If \( X \) and \( Y \) are normed spaces, if \( W \) is an open subset of \( X \) and if \( f : W \rightarrow Y \) has a directional derivative at \( \bar{x} \) in the direction \( u \), then it has a radial derivative at \( \bar{x} \) in the direction \( u \) and both derivatives coincide. In particular, if \( f \) is... | The first assertions stem from an application of the definition of a limit. | No |
Proposition 2.26. The map \( f : W \rightarrow Y \) is differentiable at \( \bar{x} \) in the direction \( u \in X \smallsetminus \) \( \{ 0\} \) if and only if \( f \) is radially differentiable at \( \bar{x} \) in the direction \( u \) and for every \( \tau > 0 \) and every (continuous) \( c : \left\lbrack {0,\tau }\... | Proof. Suppose \( f \) is differentiable at \( \bar{x} \) in the direction \( u \in X \) . Given \( \tau > 0 \) and \( c : \left\lbrack {0,\tau }\right\rbrack \rightarrow W \) that is right differentiable at 0 with \( {c}_{ + }^{\prime }\left( 0\right) = u \) and \( c\left( 0\right) = \bar{x} \), let us set \( {v}_{t} ... | Yes |
Theorem 2.28. Let \( X, Y, Z \) be normed spaces, let \( U \) and \( V \) be open subsets of \( X \) and \( Y \) respectively, and let \( f : U \rightarrow Y, g : V \rightarrow Z \) be directionally differentiable maps at \( \bar{x} \in W \mathrel{\text{:=}} {f}^{-1}\left( V\right) \) and \( \bar{y} \mathrel{\text{:=}}... | Proof. More generally, let us show that if \( f \) has a directional derivative at \( \bar{x} \) in the direction \( u \in X \) and if \( g \) has a directional derivative at \( f\left( \bar{x}\right) \) in the direction \( v \mathrel{\text{:=}} \) \( {df}\left( {\bar{x}, u}\right) \), then \( h \mathrel{\text{:=}} g \... | Yes |
Proposition 2.29. If \( f : W \rightarrow Y \) is radially differentiable at each point of a segment \( \left\lbrack {w, x}\right\rbrack \) contained in \( W \), then\n\n\[ \parallel f\left( x\right) - f\left( w\right) \parallel \leq \mathop{\sup }\limits_{{t \in \left( {0,1}\right) }}\begin{Vmatrix}{{d}_{r}f\left( {w ... | Proof. Let \( h : \left\lbrack {0,1}\right\rbrack \rightarrow Y \) be given by \( h\left( t\right) \mathrel{\text{:=}} f\left( {\left( {1 - t}\right) w + {tx}}\right) \) ; it is right differentiable on \( \left( {0,1}\right) \), with right derivative \( {h}_{ + }^{\prime }\left( t\right) = {d}_{r}f\left( {\left( {1 - t... | Yes |
Proposition 2.33. Let \( W \) be an open subset of \( X \) . If \( f : W \rightarrow Y \) is radially differentiable on a neighborhood \( V \) of \( \bar{x} \) in \( W \) and if for some \( u \in X \smallsetminus \{ 0\} \), its radial derivative \( {d}_{r}f : V \times X \rightarrow Y \) is continuous at \( \left( {\bar... | Proof. Without loss of generality, we may suppose \( u \) has norm 1 . Given \( \varepsilon > 0 \), let \( \delta \in \left( {0,1}\right) \) be such that \( \begin{Vmatrix}{{f}_{r}^{\prime }\left( {x, v}\right) - {f}_{r}^{\prime }\left( {\bar{x}, u}\right) }\end{Vmatrix} \leq \varepsilon \) for all \( \left( {x, v}\rig... | Yes |
Proposition 2.35. For every \( f \in {D}^{1}\left( {W, Y}\right) \) the map \( {f}^{\prime } : w \mapsto {Df}\left( w\right) \mathrel{\text{:=}} {df}\left( {w, \cdot }\right) \) is locally bounded. | Proof. Suppose, to the contrary, that there exist \( w \in W \) and a sequence \( \left( {w}_{n}\right) \rightarrow w \) such that \( \left( {r}_{n}\right) \mathrel{\text{:=}} \left( \begin{Vmatrix}{{Df}\left( {w}_{n}\right) }\end{Vmatrix}\right) \rightarrow + \infty \) . For each \( n \in \mathbb{N} \) one can pick so... | Yes |
Let \( f : W \rightarrow Y \) be a Hadamard (or Gâteaux) differentiable function. Then \( f \) is of class \( {D}^{1} \) if and only if \( {f}^{\prime } \) is locally bounded and for all \( u \in X \) the map \( x \mapsto {f}^{\prime }\left( x\right) u \) is continuous. In particular, if \( Y = \mathbb{R} \) and if \( ... | The necessary condition stems from the preceding proposition. The sufficient condition follows from the inequalities\n\n\[ \begin{Vmatrix}{{f}^{\prime }\left( w\right) v - {f}^{\prime }\left( x\right) u}\end{Vmatrix} \leq \begin{Vmatrix}{{f}^{\prime }\left( w\right) \left( {v - u}\right) }\end{Vmatrix} + \begin{Vmatrix... | Yes |
Proposition 2.37. If \( X, Y, Z \) are normed spaces, if \( U \) and \( V \) are open subsets of \( X \) and \( Y \) respectively, and if \( f \in {D}^{1}\left( {U, Y}\right), g \in {D}^{1}\left( {V, Z}\right) \), then \( h \mathrel{\text{:=}} g \circ f \in {D}^{1}\left( {W, Z}\right) \) for \( W \mathrel{\text{:=}} {f... | Proof. This conclusion is an immediate consequence of the formula \( {dh}\left( {w, x}\right) = \) \( {dg}\left( {f\left( w\right) ,{df}\left( {w, x}\right) }\right) \) for all \( \left( {w, x}\right) \in W \times X \) . | No |
Lemma 2.41. For all normed spaces \( W, X, Y, Z \), for every \( r \in o\left( {X, Y}\right) \) and all continuous linear maps \( A : W \rightarrow X, B : Y \rightarrow Z \) one has \( r \circ A \in o\left( {W, Y}\right) \) and \( B \circ r \in \) \( o\left( {X, Z}\right) \) (hence \( B \circ r \circ A \in o\left( {W, ... | Proof. Let \( \alpha : X \rightarrow Y \) be such that \( \alpha \left( x\right) \rightarrow 0 \) as \( x \rightarrow 0 \) and \( r\left( x\right) = \parallel x\parallel \alpha \left( x\right) \) . Then if \( A : W \rightarrow X \) is stable at 0, i.e., is such that there exists some \( c > 0 \) for which \( \parallel ... | Yes |
Proposition 2.44. If \( f : W \rightarrow Y \) is differentiable at \( \bar{x} \in W \), then it is continuous at \( \bar{x} \) . | Proof. This follows from the fact that every remainder is continuous at 0 . | No |
Proposition 2.45. If \( f, g : W \rightarrow Y \) are differentiable at \( \bar{x} \in W \), then for every \( \lambda ,\mu \in \) \( \mathbb{R} \) the map \( h \mathrel{\text{:=}} {\lambda f} + {\mu g} \) is differentiable at \( \bar{x} \) and \( {Dh}\left( \bar{x}\right) = {\lambda Df}\left( \bar{x}\right) + {\mu Dg}... | Proof. If \( r\left( x\right) \mathrel{\text{:=}} f\left( {\bar{x} + x}\right) - f\left( \bar{x}\right) - {f}^{\prime }\left( \bar{x}\right) \left( x\right), s\left( x\right) \mathrel{\text{:=}} g\left( {\bar{x} + x}\right) - g\left( \bar{x}\right) - {g}^{\prime }\left( \bar{x}\right) x \), one has \( h\left( {\bar{x} ... | Yes |
Lemma 2.46. Given an open subset \( W \) of \( X \), a map \( f : W \rightarrow Y \) is differentiable at \( \bar{x} \) if and only if there exists a map \( F : W \rightarrow L\left( {X, Y}\right) \) that is continuous at \( \bar{x} \) and such that \( f\left( x\right) - f\left( \bar{x}\right) = F\left( x\right) \left(... | Proof. Suppose there is a map \( F : W \rightarrow L\left( {X, Y}\right) \) continuous at \( \bar{x} \) such that \( f\left( x\right) = \) \( f\left( \bar{x}\right) + F\left( x\right) \left( {x - \bar{x}}\right) \) for all \( x \in W \) . Then \( f\left( x\right) - f\left( \bar{x}\right) = F\left( \bar{x}\right) \left(... | Yes |
Theorem 2.47 (Chain rule). Let \( X, Y, Z \) be normed spaces, let \( U, V \) be open subsets of \( X \) and \( Y \) respectively, and let \( f : U \rightarrow Y, g : V \rightarrow Z \) be differentiable at \( \bar{x} \in U \) and \( \bar{y} = f\left( \bar{x}\right) \) respectively and be such that \( f\left( U\right) ... | Proof. Let \( \ell \mathrel{\text{:=}} {Df}\left( \bar{x}\right), m \mathrel{\text{:=}} {Dg}\left( \bar{y}\right) \) and let \( r \in o\left( {X, Y}\right), s \in o\left( {Y, Z}\right) \) be defined by\n\n\[ \nr\left( x\right) \mathrel{\text{:=}} f\left( {\bar{x} + x}\right) - f\left( \bar{x}\right) - \ell \left( x\rig... | Yes |
Corollary 2.49. The differentiability of \( f : W \rightarrow Y \) (with \( W \) open in \( X \) ) at \( \bar{x} \) does not depend on the choices of the norms on \( X \) and \( Y \) within their equivalences classes. | In fact, changing the norm amounts to composing with a continuous linear map. | No |
Let \( X, Y \) be normed spaces, let \( W \) be an open subset of \( X \), and let \( f : W \rightarrow Y \) . If \( f \) is Fréchet differentiable at \( \bar{x} \in W \), then \( f \) is Hadamard differentiable at \( \bar{x} \) . If \( X \) is finite-dimensional, the converse holds. | The first assertion follows from the definitions or from Theorem 2.47 and Proposition 2.26. | No |
Proposition 2.51. If \( f \) is Gâteaux differentiable on \( W \) and if \( {f}^{\prime } : W \rightarrow L\left( {X, Y}\right) \) is continuous at \( \bar{x} \in W \), then \( f \) is Fréchet differentiable at \( \bar{x} \) . | Proof. Without loss of generality, replacing \( Y \) by its completion, we may suppose \( Y \) is complete; replacing \( W \) by a ball centered at \( \bar{x} \), we may also suppose \( W \) is convex. Then for \( x \in W \) one has \( f\left( x\right) - f\left( \bar{x}\right) = F\left( x\right) \left( {x - \bar{x}}\ri... | Yes |
Let \( X,{Y}_{1},\ldots ,{Y}_{n} \) be normed spaces, let \( W \) be an open subset of \( X \), and let \( f \mathrel{\text{:=}} \left( {{f}_{1},\ldots ,{f}_{n}}\right) : W \rightarrow Y \mathrel{\text{:=}} {Y}_{1} \times \cdots \times {Y}_{n} \) . Then \( f \) is differentiable at \( \bar{x} \in W \) if and only if it... | Proof. Let \( {p}_{i} : Y \rightarrow {Y}_{i} \) denote the \( i \) th canonical projection. If \( f \) is differentiable at \( \bar{x} \) , then Corollary 2.48 ensures that \( {f}_{i} \mathrel{\text{:=}} {p}_{i} \circ f \) is differentiable at \( \bar{x} \) and \( D{f}_{i}\left( \bar{x}\right) = {p}_{i} \circ \) \( {D... | Yes |
Proposition 2.56. Let \( X \) and \( Y \) be normed spaces, let \( W \) be an open subset of \( X \) and let \( \bar{x} \in W \) . A map \( f : W \rightarrow Y \) that is differentiable on a neighborhood \( U \subset W \) of \( \bar{x} \) is circa-differentiable at \( \bar{x} \in W \) if and only if \( f \in {C}_{\bar{... | Proof. Suppose \( f \in {C}_{\bar{x}}^{1}\left( {W, Y}\right) \) and let \( \ell \mathrel{\text{:=}} {Df}\left( \bar{x}\right) \) . Given \( \varepsilon > 0 \) one can find \( \delta > 0 \) such that \( B\left( {\bar{x},\delta }\right) \subset W \) and for \( x \in B\left( {\bar{x},\delta }\right) \) one has \( \parall... | Yes |
Proposition 2.57. If \( f : W \rightarrow Y \) is defined on an open subset \( W \) of a product space \( X \mathrel{\text{:=}} {X}_{1} \times \cdots \times {X}_{k} \), if for \( i = 1,\ldots, k \) , \( f \) has a partial derivative at \( \bar{x} \in W \) relative to \( {X}_{i} \) , and if \( f \) is circa-differentiab... | Proof. It suffices to give the proof for \( k = 2 \) ; an induction yields the general case.\n\nThus, let \( f \) be circa-differentiable at \( \bar{x} \) with respect to \( {X}_{1} \) and have a partial derivative at \( \bar{x} \) relative to \( {X}_{2} \) . The first assumption means that there exists some \( {\ell }... | Yes |
Let \( X, Y \) be normed spaces, \( Y \) being complete, and let \( W \) be an open subset of \( X \) . The space \( {B}^{1}\left( {W, Y}\right) \) (resp. \( B{C}^{1}\left( {W, Y}\right) \) ) of bounded, Lipschitzian, differentiable (resp. of class \( {C}^{1} \) ) maps from \( W \) to \( Y \) is complete for the norm \... | Proof. Let \( \left( {f}_{n}\right) \) be a Cauchy sequence of \( \left( {{B}^{1}\left( {W, Y}\right) ,\parallel \cdot {\parallel }_{1,\infty }}\right) \) . Then \( \left( {f}_{n}^{\prime }\right) \) is a Cauchy sequence of the space \( B\left( {W, L\left( {X, Y}\right) }\right) \) of bounded maps from \( W \) into \( ... | Yes |
Theorem 2.62 (Borwein-Preiss variational principle). Let \( X \) be a Banach space and let \( F \mathrel{\text{:=}} {B}^{1}\left( X\right) \) (resp. \( B{H}^{1}\left( X\right), B{C}^{1}\left( X\right), B{D}^{1}\left( X\right) \) ) with the norm \( \parallel \cdot {\parallel }_{1,\infty } \) defined above. Suppose there... | Proof. Conditions (b) and (c) of Theorem 1.152 are obviously satisfied, whereas (a) is part of our assumptions (here we have changed \( W \) into \( F \) in order to avoid confusion with what precedes). Moreover, \( \left( {F,\parallel \cdot \parallel }\right) \) is complete by the preceding corollary. The last asserti... | Yes |
Lemma 2.64. The following assertions are equivalent:\n\n(a) \( f \) has a Newton approximation A that is bounded near \( \bar{x} \)\n\n(b) \( f \) has a slant derivative A at \( \bar{x} \) that is bounded on some neighborhood of \( \bar{x} \)\n\n(c) \( f \) is stable at \( \bar{x} \), i.e., there exist \( c > 0, r > 0 ... | Proof. (a) \( \Rightarrow \) (c) If for some \( \alpha ,\beta > 0 \) and some \( r > 0 \) a map \( A : B\left( {\bar{x}, r}\right) \rightarrow L\left( {X, Y}\right) \) is such that (2.15) holds with \( \parallel A\left( x\right) \parallel \leq \beta \) for all \( x \in B\left( {\bar{x}, r}\right) \), then by the triang... | Yes |
Proposition 2.65. Let \( \bar{x} \) be a solution to (2.14), let \( \alpha ,\beta, r > 0 \) satisfy \( \gamma \mathrel{\text{:=}} {\alpha \beta } < 1 \) , and let \( A : B\left( {\bar{x}, r}\right) \rightarrow L\left( {X, Y}\right) \) be such that (2.15) holds, \( A\left( x\right) \) being invertible with \( \begin{Vma... | Proof. Using the fact that \( f\left( \bar{x}\right) = 0 \), so that\n\n\[ \n{x}_{n + 1} - \bar{x} = A{\left( {x}_{n}\right) }^{-1}\left( {f\left( \bar{x}\right) - f\left( {x}_{n}\right) + A\left( {x}_{n}\right) \left( {{x}_{n} - \bar{x}}\right) }\right) , \n\]\n\nwe inductively obtain that\n\n\[ \n\begin{Vmatrix}{{x}_... | Yes |
Theorem 2.66 (Kantorovich). Let \( {x}_{0} \in W,\alpha ,\beta > 0, r > 0 \) with \( \gamma \mathrel{\text{:=}} {\alpha \beta } < 1 \) , \( B\left( {{x}_{0}, r}\right) \subset W \) and let \( A : B\left( {{x}_{0}, r}\right) \rightarrow L\left( {X, Y}\right) \) be such that for all \( x \in B\left( {{x}_{0}, r}\right) \... | Proof. Let us prove by induction that \( {x}_{n} \in B\left( {{x}_{0}, r}\right) ,\begin{Vmatrix}{{x}_{n + 1} - {x}_{n}}\end{Vmatrix} \leq \beta {\gamma }^{n}\begin{Vmatrix}{f\left( {x}_{0}\right) }\end{Vmatrix} \), and \( \begin{Vmatrix}{f\left( {x}_{n}\right) }\end{Vmatrix} \leq {\gamma }^{n}\begin{Vmatrix}{f\left( {... | Yes |
Theorem 2.67 (Lyusternik-Graves theorem). Let \( X \) and \( Y \) be Banach spaces, let \( W \) be an open subset of \( X \), and let \( g : W \rightarrow Y \) be circa-differentiable at some \( \bar{x} \in W \) with a surjective derivative \( {Dg}\left( \bar{x}\right) \) . Then \( g \) is open at \( \bar{x} \) . More ... | Proof. Let \( A : W \rightarrow L\left( {X, Y}\right) \) be the constant map with value \( A \mathrel{\text{:=}} {Dg}\left( \bar{x}\right) \) (we use a familiar abuse of notation). The open mapping theorem yields some \( \beta > 0 \) and some right inverse \( B : Y \rightarrow X \) of \( A \) such that \( \parallel B\l... | Yes |
Proposition 2.68. Let \( f \) be a linear isomorphism between two Banach spaces \( X \) and \( Y \) . Then every \( g \in L\left( {X, Y}\right) \) such that \( \parallel f - g\parallel < {\begin{Vmatrix}{f}^{-1}\end{Vmatrix}}^{-1} \) is an isomorphism. | Proof. Let us first consider the case \( X = Y, f = {I}_{X} \) . Let \( u \mathrel{\text{:=}} {I}_{X} - g \), so that \( u \in \) \( L\left( {X, X}\right) \) satisfies \( \parallel u\parallel < 1 \) . Since the map \( \left( {v, w}\right) \mapsto w \circ v \) is continuous, since\n\n\[ \n{I}_{X} - {u}^{n + 1} = \left( ... | Yes |
Lemma 2.69. Let \( U \) and \( V \) be two open subsets of normed spaces \( X \) and \( Y \) respectively. Assume that \( f : U \rightarrow V \) is a homeomorphism that is differentiable at \( a \in U \) and such that \( {f}^{\prime }\left( a\right) \) is an isomorphism. Then the inverse \( g \) of \( f \) is different... | Proof. Using translations if necessary, we may suppose \( a = 0, f\left( a\right) = 0 \) without loss of generality. Changing \( f \) into \( {h}^{-1} \circ f \), where \( h \mathrel{\text{:=}} {f}^{\prime }\left( a\right) \), we may also suppose \( Y = X \) and \( {f}^{\prime }\left( a\right) = {I}_{X} \) . Then setti... | Yes |
Lemma 2.70. Let \( \left( {U, d}\right) \) be a metric space, let \( Y \) be a normed space, let \( j, h : U \rightarrow Y \) be such that\n\n(a) \( j \) is injective and its inverse \( {j}^{-1} : j\left( U\right) \rightarrow U \) is Lipschitzian with rate \( \gamma \) ;\n\n(b) \( h \) is Lipschitzian with rate \( \lam... | Proof. The lemma results from the following relations, valid for every \( x,{x}^{\prime } \in X \) :\n\n\[ \begin{Vmatrix}{g\left( x\right) - g\left( {x}^{\prime }\right) }\end{Vmatrix} \geq \begin{Vmatrix}{e\left( x\right) - e\left( {x}^{\prime }\right) }\end{Vmatrix} - \begin{Vmatrix}{h\left( x\right) - h\left( {x}^{... | Yes |
Lemma 2.71. Let \( W \) be an open subset of a Banach space \( Y \) and let \( k : W \rightarrow Y \) be a Lipschitzian map with rate \( c < 1 \) . Then the image of \( W \) by \( f \mathrel{\text{:=}} {I}_{W} + k \) is open. | Proof. We will prove that for every \( a \in W \) and for every closed ball \( B\left\lbrack {a, r}\right\rbrack \) contained in \( W \), the closed ball \( B\left\lbrack {f\left( a\right) ,\left( {1 - c}\right) r}\right\rbrack \) is contained in the set \( f\left( W\right) \), and in fact in the set \( f\left( {B\left... | Yes |
Lemma 2.72. Let \( \left( {U, d}\right) \) be a metric space, let \( Y \) be a Banach space, let \( \gamma > 0,\lambda > 0 \) with \( {\gamma \lambda } < 1 \), and let \( j, h : U \rightarrow Y \) be such that \( W \mathrel{\text{:=}} j\left( U\right) \) is open and\n\n(a) \( j \) is injective and its inverse \( {j}^{-... | Proof. Let \( k \mathrel{\text{:=}} h \circ {j}^{-1} \), so that \( f \circ {j}^{-1} = {I}_{W} + k \) and \( k \) is Lipschitzian with rate \( {\gamma \lambda } < 1 \) . Then Lemma 2.71 shows that \( f\left( U\right) = f\left( {{j}^{-1}\left( W\right) }\right) = \left( {I + k}\right) \left( W\right) \) is open. | Yes |
Theorem 2.73 (Inverse mapping theorem). Let \( X \) and \( Y \) be Banach spaces, let \( W \) be an open subset of \( X \), and let \( f : W \rightarrow Y \) be circa-differentiable at \( a \in W \) and such that \( {f}^{\prime }\left( a\right) \) is an isomorphism from \( X \) onto \( Y \) . Then there exist neighborh... | Proof. In the preceding lemma, let us take \( j \mathrel{\text{:=}} {f}^{\prime }\left( a\right), h = f - j \) . Since \( j \) is an isomorphism, its inverse is Lipschitzian with rate \( \begin{Vmatrix}{j}^{-1}\end{Vmatrix} \) . Let \( U \) be a neighborhood of \( a \) such that \( h \) is Lipschitzian with rate \( \la... | Yes |
Lemma 2.75. Let \( X \) and \( Y \) be Banach spaces. Then the set \( \operatorname{Iso}\left( {X, Y}\right) \) of isomorphisms from \( X \) onto \( Y \) is open in \( L\left( {X, Y}\right) \) and the map \( i : \operatorname{Iso}\left( {X, Y}\right) \rightarrow \operatorname{Iso}\left( {Y, X}\right) \) given by \( i\l... | Proof. The first assertion has been proved in Proposition 2.68. Let us prove the second assertion by first considering the case \( X = Y \) and by showing that \( i \) is differentiable at the identity map \( {I}_{X} \), with derivative \( {Di}\left( {I}_{X}\right) \) given by \( {Di}\left( {I}_{X}\right) \left( v\righ... | Yes |
Corollary 2.76. Let \( X \) and \( Y \) be Banach spaces, let \( W \) be an open subset of \( X \), and let \( f : W \rightarrow Y \) be of class \( {C}^{k}\left( {k \geq 1}\right) \) and such that \( {f}^{\prime }\left( a\right) \) is an isomorphism from \( X \) onto \( Y \) for some \( a \in W \) . Then there exist n... | Proof. Let us first consider the case \( k = 1 \) . The inverse mapping theorem ensures that \( f \) induces a homeomorphism from a neighborhood \( U \) of \( a \) onto a neighborhood \( V \) of \( b \) . Since \( {f}^{\prime } \) is continuous at \( a \) and since the set \( \operatorname{Iso}\left( {X, Y}\right) \) o... | Yes |
Corollary 2.77. Let \( X \) and \( Y \) be Banach spaces, let \( W \) be an open subset of \( X \), and let \( f : W \rightarrow Y \) be an injection of class \( {C}^{k} \) such that for every \( x \in W \), the linear map \( {f}^{\prime }\left( x\right) \) is an isomorphism from \( X \) onto \( Y \). Then \( f\left( W... | Proof. The inverse mapping theorem ensures that \( f\left( W\right) \) is open in \( Y \). Thus \( f \) is a continuous bijection from \( W \) onto \( f\left( W\right) \) and its inverse is locally of class \( {C}^{k} \), hence is of class \( {C}^{k} \). | Yes |
Theorem 2.78. Let \( X, Y, Z \) be Banach spaces, let \( W \) be an open subset of \( X \times Y \) , and let \( f : W \rightarrow Z \) be a map of class \( {C}^{1} \) at \( \left( {a, b}\right) \in W \) such that \( f\left( {a, b}\right) = 0 \) and the second partial derivative \( {D}_{Y}f\left( {a, b}\right) \) is an... | Proof. Let \( F : W \rightarrow X \times Z \) be the map given by \( F\left( {x, y}\right) \mathrel{\text{:=}} \left( {x, f\left( {x, y}\right) }\right) \) . Then \( F \) is of class \( {C}^{1} \) at \( \left( {a, b}\right) \), as are its components, and\n\n\[ {DF}\left( {a, b}\right) \left( {x, y}\right) = \left( {x,{... | Yes |
Theorem 2.79. Let \( X, Y, Z \) be Banach spaces, \( Y \) and \( Z \) being finite-dimensional, let \( W \) be an open subset of \( X \times Y \), and let \( f : W \rightarrow Z \) be Fréchet differentiable at \( \left( {a, b}\right) \in \) \( W \) such that \( f\left( {a, b}\right) = 0 \) and the partial derivative \(... | Proof. Using translations and composing \( f \) with \( {D}_{Y}f{\left( a, b\right) }^{-1} \), we may suppose \( \left( {a, b}\right) = \left( {0,0}\right), Z = Y \), and \( {D}_{Y}f\left( {a, b}\right) = {I}_{Y} \) . Let \( r : W \rightarrow Y \) be a remainder such that\n\n\[ f\left( {x, y}\right) \mathrel{\text{:=}}... | Yes |
Theorem 2.81. Let \( X, Y, Z \) be Banach spaces, \( Y \) and \( Z \) being finite-dimensional, let \( W \) be an open subset of \( X \times Y \), and let \( f : W \rightarrow Z \) be a map of class \( {D}^{1} \) at \( \left( {a, b}\right) \in W \) such that \( f\left( {a, b}\right) = 0 \) and the partial derivative \(... | Proof. We may suppose \( W \) is a ball \( B\left( {\left( {a, b}\right) ,{\rho }_{0}}\right), Y = Z,{D}_{Y}f\left( {a, b}\right) = {I}_{Y} \) . With the notation of the preceding proof, using the compactness of the unit ball of \( Y \), we may suppose the remainder \( r \) satisfies, for \( \rho \in \left( {0,{\rho }_... | Yes |
Lemma 2.83. If \( f \) is a Legendre function on \( U \), then its Legendre transform \( {f}^{L} \) is of class \( {C}^{1} \) on \( V \mathrel{\text{:=}} {f}^{\prime }\left( U\right) \) and of class \( {C}^{k}\left( {k \geq 1}\right) \) if \( f \) is of class \( {C}^{k} \) . Moreover, \( {f}^{L} \) is a Legendre functi... | Proof. Given \( v \mathrel{\text{:=}} {Df}\left( u\right) \in V \), let \( y \in V - v \), let \( x \mathrel{\text{:=}} h\left( {v + y}\right) - h\left( v\right) \in U - u \), and let \( r\left( x\right) = f\left( {u + x}\right) - f\left( u\right) - {Df}\left( u\right) x \) . Then since \( h\left( v\right) = u, h\left(... | Yes |
Proposition 2.88. If \( S \) is \( {C}^{1} \) -smooth around \( a \in S \), then the tangent cone \( T\left( {S, a}\right) \) to \( S \) at a coincides with the set \( {T}^{I}\left( {S, a}\right) \) of \( v \in X \) such that there exist \( \tau > 0 \) and \( c : \left\lbrack {0,\tau }\right\rbrack \rightarrow X \) rig... | Proof. The result follows from Lemma 2.87 and the observation that if \( S \) is an open subset of some closed linear subspace \( L \) of \( X \) then \( T\left( {S, a}\right) = L = {T}^{I}\left( {S, a}\right) \) . | No |
Theorem 2.89 (Submersion theorem). Let \( X \) and \( Z \) be Banach spaces, let \( W \) be an open subset of \( X \), and let \( g : W \rightarrow Z \) be a map of class \( {C}^{k} \) with \( k \geq 1 \) such that for some \( a \in W \) the map \( {Dg}\left( a\right) \) is surjective and its kernel \( N \) has a topol... | Proof. Let \( F : W \rightarrow N \times Z \) be given by \( F\left( x\right) = \left( {{p}_{N}\left( x\right) - {p}_{N}\left( a\right), g\left( x\right) }\right) \), where \( {p}_{N} : X \rightarrow N \) is the projection on \( N \) associated with the isomorphism between \( X \) and \( M \times N \) . Then \( F \) is... | Yes |
Corollary 2.90. Let \( X \) and \( Z \) be Banach spaces, let \( W \) be an open subset of \( X \), and let \( g : W \rightarrow Z \) be a map of class \( {C}^{k} \) with \( k \geq 1 \) . Let\n\n\[ S \mathrel{\text{:=}} \{ x \in W : g\left( x\right) = 0\} . \]\n\nSuppose that for some \( a \in S \) the map \( {g}^{\pri... | Proof. Using the notation of the submersion theorem, setting \( Y \mathrel{\text{:=}} N \), we see that Definition 2.84 is satisfied, noting that for \( x \in U \) we have \( x \in S \cap U \) iff \( p\left( {\varphi \left( x\right) }\right) = \) \( g\left( x\right) = 0 \), iff \( \varphi \left( x\right) \in \left( {Y\... | Yes |
Proposition 2.91 (Lyusternik). Let \( X \) and \( Y \) be Banach spaces, let \( W \) be an open subset of \( X \), and let \( g : W \rightarrow Y \) be circa-differentiable at \( a \in S \mathrel{\text{:=}} \{ x \in W : g\left( x\right) = 0\} \) , with \( {g}^{\prime }\left( a\right) \left( X\right) = Y \) . Then \( T\... | Proof. The inclusion \( T\left( {S, a}\right) \subset \ker {g}^{\prime }\left( a\right) \) follows from Lemma 2.87. Conversely, let \( v \in \ker {g}^{\prime }\left( a\right) \) . Theorem 2.67 yields some \( \kappa ,\rho > 0 \) such that for all \( w \in B\left( {a,\rho }\right) \) there exists some \( x \in W \) such ... | Yes |
Theorem 2.93 (Immersion theorem). Let \( P \) and \( X \) be Banach spaces, let \( O \) be an open subset of \( P \), and let \( f : O \rightarrow X \) be a map of class \( {C}^{k} \) with \( k \geq 1 \) such that for some \( \bar{p} \in O \) the map \( {Df}\left( \bar{p}\right) \) is injective and its image \( Y \) ha... | Proof. Let \( F : O \times Z \rightarrow X \) be given by \( F\left( {p, z}\right) = f\left( p\right) + z \) . Then \( F \) is of class \( {C}^{k} \) and \( {F}^{\prime }\left( {\bar{p},0}\right) \left( {p, z}\right) = {f}^{\prime }\left( \bar{p}\right) \left( p\right) + z \) for \( \left( {p, z}\right) \in P \times Z ... | Yes |
Corollary 2.94 (Embedding theorem). Let \( P \) and \( X \) be Banach spaces, let \( O \) be an open subset of \( P \), and let \( f : O \rightarrow X \) be a map of class \( {C}^{k} \) with \( k \geq 1 \) such that for every \( p \in O \) the map \( {f}^{\prime }\left( p\right) \) is injective and its image has a topo... | Proof. Given \( a \mathrel{\text{:=}} f\left( p\right) \) in \( S \), with \( p \in O \), we take \( {Q}_{a} \subset O,{U}_{a} \subset X,{W}_{a} \subset Z \) and a \( {C}^{k} \) -diffeomorphism \( {\psi }_{a} : {V}_{a} \mathrel{\text{:=}} {Q}_{a} \times {W}_{a} \rightarrow {U}_{a} \) such that \( {\psi }_{a}\left( {q,0... | Yes |
Theorem 2.97 (Fermat’s rule). Suppose \( f \) attains a local maximum on \( F \) at \( \bar{x} \) and is Fréchet differentiable at \( \bar{x} \). Then\n\n\[ \n{f}^{\prime }\left( \bar{x}\right) \in {N}_{F}\left( {F,\bar{x}}\right) \n\]\n\nIf \( f \) attains a local minimum on \( F \) at \( \bar{x} \) and is Fréchet dif... | Proof. Suppose \( f \) attains a local maximum on \( F \) at \( \bar{x} \) and is differentiable at \( \bar{x} \). Set\n\n\[ \nf\left( x\right) = f\left( \bar{x}\right) + \left\langle {{\bar{x}}^{ * }, x - \bar{x}}\right\rangle + r\left( {x - \bar{x}}\right) \n\]\n\nwith \( r \) a remainder, \( {\bar{x}}^{ * } \mathrel... | Yes |
Proposition 2.100. The normal cone to \( F \) at \( \bar{x} \) is the polar cone to the tangent cone to \( F \) at \( \bar{x} \) : | Proof. Given \( {\bar{x}}^{ * } \in N\left( {F,\bar{x}}\right) \) and \( u \in T\left( {F,\bar{x}}\right) \smallsetminus \{ 0\} \), for every \( \varepsilon > 0 \), taking \( \delta \in \left( {0,\varepsilon }\right) \) such that \( \left\langle {{\bar{x}}^{ * }, v}\right\rangle \leq \varepsilon \) for every \( \left( ... | Yes |
Theorem 2.101 (Fermat’s rule). Suppose \( f \) attains a local maximum on \( F \) at \( \bar{x} \in F \) and is Hadamard differentiable at \( \bar{x} \) . Then for all \( v \in T\left( {F,\bar{x}}\right) \) one has \( {f}^{\prime }\left( \bar{x}\right) v \leq 0 \) : | Proof. Let \( V \) be an open neighborhood of \( \bar{x} \) in \( X \) such that \( f\left( x\right) \leq f\left( \bar{x}\right) \) for all \( x \in F \cap \) \( V \) . Given \( v \in T\left( {F,\bar{x}}\right) \), let \( \left( {v}_{n}\right) \rightarrow v,\left( {t}_{n}\right) \rightarrow {0}_{ + } \) be sequences su... | Yes |
Theorem 2.102. Suppose \( f \) attains a local maximum on \( F \) at \( \bar{x} \) . Then\n\n\[ \n{f}^{D}\left( {\bar{x}, u}\right) \leq 0\text{ for all }u \in T\left( {F,\bar{x}}\right) .\n\] | Proof. Let \( u \in T\left( {F,\bar{x}}\right) \) . There exist \( \left( {t}_{n}\right) \rightarrow {0}_{ + },\left( {u}_{n}\right) \rightarrow u \) such that \( \bar{x} + {t}_{n}{u}_{n} \in F \) for all \( n \in \mathbb{N} \) . For \( n \) large enough we have \( f\left( {\bar{x} + {t}_{n}{u}_{n}}\right) \leq f\left(... | Yes |
Proposition 2.104. Suppose \( f \) is directionally stable at \( \bar{x} \) in the sense that for all \( u \in X \smallsetminus \{ 0\} \) one has \( \left( {1/t}\right) \left( {f\left( {\bar{x} + {tv}}\right) - f\left( {\bar{x} + {tu}}\right) }\right) \rightarrow 0 \) as \( \left( {t, v}\right) \rightarrow \left( {0, u... | Proof. Suppose, to the contrary, that there exists some \( u \in T\left( {F,\bar{x}}\right) \) such that \( {f}^{I}\left( {\bar{x}, u}\right) < 0 \) . Then there exists some \( r < 0 \) such that \( \left( {u, r}\right) \in {T}^{I}\left( {{E}_{f},{\bar{x}}_{f}}\right) \) ; thus, if \( \left( {t}_{n}\right) \rightarrow ... | Yes |
Corollary 2.109. Let \( g : U \rightarrow V \) be a bijection between two open subsets of the normed spaces \( X \) and \( Y \) respectively such that \( g \) and \( h \mathrel{\text{:=}} {g}^{-1} \) are \( H \) -differentiable, respectively \( F \) -differentiable, at \( \bar{x} \) and \( \bar{y} \mathrel{\text{:=}} g... | Proof. Since \( {h}^{\prime }{\left( \bar{y}\right) }^{\top } \) is the inverse of \( {g}^{\prime }{\left( \bar{x}\right) }^{\top } \), one has the inclusions of Proposition 2.108 and their analogues in which \( h,\bar{y}, C \) take the roles of \( g,\bar{x}, B \) , respectively. | Yes |
Proposition 2.110 (Lyusternik). Let \( X, Y \) be Banach spaces, let \( U \) be an open subset of \( X \), and let \( g : U \rightarrow Y \) be circa-differentiable at \( \bar{x} \in U \) with \( {g}^{\prime }\left( \bar{x}\right) \left( X\right) = Y \) . Then for \( S \mathrel{\text{:=}} {g}^{-1}\left( \bar{y}\right) ... | Proof. Proposition 2.108 ensures that \( {g}^{\prime }{\left( \bar{x}\right) }^{\top }\left( {Y}^{ * }\right) \subset {N}_{F}\left( {S,\bar{x}}\right) \subset N\left( {S,\bar{x}}\right) \) . Now, given \( {x}^{ * } \in N\left( {S,\bar{x}}\right) \), for all \( v \in T\left( {S,\bar{x}}\right) = \ker {g}^{\prime }\left(... | Yes |
Theorem 2.111. Let \( X, Y \) be Banach spaces, let \( U \) be an open subset of \( X \), and let \( g : U \rightarrow Y \) be a map that is circa-differentiable at \( \bar{x} \in U \) with \( A \mathrel{\text{:=}} {g}^{\prime }\left( \bar{x}\right) \) surjective. Then if \( C \) is a subset of \( Y \) and if \( \bar{x... | Proof. We prove the Fréchet case only, leaving the directional case to the reader. The Lyusternik-Graves theorem (Theorem 2.67) asserts the existence of \( \sigma > 0, c > \) 0 such that for all \( y \in B\left( {\bar{y},\sigma }\right) \) there exists \( {x}_{y} \in {g}^{-1}\left( y\right) \) satisfying \( \begin{Vmat... | No |
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