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Proposition 15.6 Let \( \lambda \in \mathbb{C} \smallsetminus \mathbb{R} \) . Then for each end point of the interval \( \left( {a, b}\right) \), there exists a nonzero solution of (15.5) which is in \( {L}^{2} \) near to it. | Proof We carry out the proof for the end point \( b \) ; the case of \( a \) is similar.\n\nFix \( c \in \left( {a, b}\right) \) and let \( {T}_{c} \) denote the corresponding operator on the interval \( \left( {c, b}\right) \) in \( {L}^{2}\left( {c, b}\right) \) . Choosing functions \( f, g \in {C}_{0}^{\infty }\left... | Yes |
Corollary 15.7 If at least one end point is regular, then the deficiency indices of \( T \) are \( \left( {1,1}\right) \) or \( \left( {2,2}\right) \) . If both end points are regular, then \( T \) has deficiency indices \( \left( {2,2}\right) \) . | Proof Let \( \lambda \in \mathbb{C} \smallsetminus \mathbb{R} \) . If both end points are regular, by Proposition 15.5(ii) all solutions \( f \) of (15.5) are in \( {L}^{2} \) near \( a \) and in \( {L}^{2} \) near \( b \), so \( f \in \mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) \) . Thus, \( \dim \mathcal{N}\le... | Yes |
Example 15.1 \( \left( {q\left( x\right) \equiv 0\text{on}\left( {a, b}\right) }\right) \) For \( \lambda \in \mathbb{C},\lambda \neq 0 \), a fundamental system of Eq. (15.5) is given by \[ \left\{ {{u}_{1}\left( {x;\lambda }\right) \mathrel{\text{:=}} \sin \sqrt{\lambda }x,{u}_{2}\left( {x;\lambda }\right) \mathrel{\t... | If \( a \) or \( b \) is in \( \mathbb{R} \), it is a regular end point, and \( T \) is in the limit circle case at \( a \) resp. \( b \) . If \( a = - \infty \) or \( b = \infty \), then \( {u}_{1}\left( {\cdot ;\lambda }\right) \notin {L}^{2}\left( {a, b}\right) \), so \( T \) is in the limit point case. | No |
Lemma 15.9 If \( T \) is in the limit point case at a (resp. b), then \( {\left\lbrack f, g\right\rbrack }_{a} = 0 \) (resp. \( \left. {{\left\lbrack f, g\right\rbrack }_{b} = 0}\right) \) for all \( f, g \in \mathcal{D}\left( {T}^{ * }\right) \) . | Proof We prove this for the end point \( b \) . As in the proof of Proposition 15.6, we fix \( c \in \left( {a, b}\right) \) and consider the operator \( {T}_{c} \) on the interval \( \left( {c, b}\right) \) in \( {L}^{2}\left( {c, b}\right) \) . Let \( \lambda \in \mathbb{C} \smallsetminus \mathbb{R} \) . Since \( \ma... | Yes |
Proposition 15.11 Let \( b = + \infty \) and suppose that \( q \) is bounded from below near to \( + \infty \) (that is, there are numbers \( c > a \) and \( \gamma \in \mathbb{R} \) such that \( q\left( x\right) \geq \gamma \) for all \( x \geq c \) ). Then \( T \) is in the limit point case at \( b = + \infty \) . | Proof By Proposition 15.3(ii), there is a unique solution \( u \) of the differential equation \( \mathcal{L}\left( f\right) \equiv - {f}^{\prime \prime } + {qf} = {\gamma f} \) on \( \left( {a, + \infty }\right) \) satisfying \( u\left( c\right) = 1,{u}^{\prime }\left( c\right) = 0 \) . Since \( \bar{u} \) is a soluti... | Yes |
Proposition 15.12 Suppose that \( a = 0 \) and \( b \in (0, + \infty \rbrack \) .\n\n(i) If there exists a positive number \( c \) such that \( c < b \) and \( q\left( x\right) \geq \frac{3}{4}{x}^{-2} \) for all \( x \in \left( {0, c}\right) \), then \( T \) is in the limit point case at \( a = 0 \) . | Proof (i): Put \( {u}_{0}\left( x\right) \mathrel{\text{:=}} {x}^{-1/2} \) . Clearly, the function \( {u}_{0} \) is not in \( {L}^{2} \) near \( a = 0 \), and it satisfies the equation \( - {u}_{0}^{\prime \prime }\left( x\right) + \frac{3}{4}{x}^{-2}{u}_{0}\left( x\right) = 0 \) on \( \left( {0, b}\right) \) .\n\nBy P... | Yes |
Example 15.2 \( \left( {q\left( x\right) = \gamma {x}^{-2},\gamma \in \mathbb{R}\text{, on}\left( {0, + \infty }\right) }\right) \) Then \( \mathcal{L}\left( f\right) = - {f}^{\prime \prime } + {qf} = 0 \) is one form of Bessel’s equation. The preceding criterion implies that \( T \) is in the limit point case at \( b ... | By Proposition 15.12, \( T \) is in the limit point case at \( a = 0 \) if and only if \( \gamma \geq 3/4 \) . Therefore, by Theorem 15.10, \( T \) is essentially self-adjoint if and only if \( \gamma \geq 3/4 \) . Otherwise, the symmetric operator \( T \) has deficiency indices \( \left( {1,1}\right) \) . | No |
Example 15.3 (Regular end points \( a \) and \( b \) ) Throughout this example we assume that both end points \( a \) and \( b \) are regular. Recall that by Definition 15.1 this holds if and only if \( a \) and \( b \) are in \( \mathbb{R} \) and \( q \in {L}^{1}\left( {a, b}\right) \) . | Then the boundary triplet in Example 14.2 for \( q = 0 \) generalizes verbatim to the present case. Indeed, if \( f \in \mathcal{D}\left( {T}^{ * }\right) \), then \( f \) and \( {f}^{\prime } \) can be extended to continuous functions on \( \left\lbrack {a, b}\right\rbrack \) by Proposition 15.5(i). Therefore, by comb... | Yes |
In this example we suppose that \( T \) is in the limit circle case at both end points \( a \) and \( b \). In order to construct a boundary triplet for \( {T}^{ * } \), we assume that \( {u}_{1} \) and \( {u}_{2} \) are real-valued functions of \( \mathcal{D}\left( {T}^{ * }\right) \) satisfying\n\n\[ \n{\left\lbrack ... | Let \( c \in \left( {a, b}\right) \) and \( \lambda \in \mathbb{R} \). By Proposition 15.3(ii) there exist unique solutions \( {u}_{1},{u}_{2} \) of the differential equation \( - {u}^{\prime \prime } + {qu} = {\lambda u} \) on \( \left( {a, b}\right) \) satisfying \( {u}_{1}\left( c\right) = {u}_{2}^{\prime }\left( c\... | Yes |
Assume now that the end point \( a \) is regular and \( T \) is in the limit point case at \( b \). | Then, by Proposition 15.5(i) and Lemma 15.9, \( f \) and \( {f}^{\prime } \) extend to continuous functions on \( \lbrack a, b) \), and we have \( {\left\lbrack f, g\right\rbrack }_{b} = 0 \) for \( f, g \in \mathcal{D}\left( {T}^{ * }\right) \) . Therefore, by (15.4), \[ {\left\lbrack f, g\right\rbrack }_{{T}^{ * }} =... | Yes |
Example 15.6 (Example 15.1 continued: \( q\left( x\right) \equiv 0 \) on \( \left( {0, + \infty }\right) \) ) Then \( a = 0 \) is a regular end point, and \( T \) is in the limit point case at \( b = + \infty \) . The fundamental system from (15.16) is given by \( \mathrm{s}\left( {x;0}\right) = x,\mathrm{c}\left( {x;0... | \[ \mathrm{s}\left( {x;z}\right) = \frac{1}{\sqrt{z}}\sin \sqrt{z}x,\;\mathrm{c}\left( {x;z}\right) = \cos \sqrt{z}x,\;z \in \mathbb{C}, z \neq 0. \] For \( z \in \mathbb{C} \smallsetminus \lbrack 0,\infty ) \), let \( \sqrt{z} \) denote the square root of \( z \) satisfying \( \operatorname{Im}\sqrt{z} > 0 \) . Then \... | Yes |
Example 15.7 (Limit circle case at a and limit point case at \( b \) ) Suppose that \( T \) is in the limit circle case at \( a \) and in the limit point case at \( b \) .\n\nLet us assume that \( {u}_{1} \) and \( {u}_{2} \) are real-valued functions from \( \mathcal{D}\left( {T}^{ * }\right) \) such that\n\n\[{\left\... | To show that such functions exist, we modify the reasoning from Example 15.4. Let us denote the functions \( {u}_{1},{u}_{2} \) constructed therein by \( {\widetilde{u}}_{1},{\widetilde{u}}_{2} \), respectively. Since \( T \) is in the limit point case at \( b \), we cannot conclude that \( {\widetilde{u}}_{1},{\wideti... | Yes |
Proposition 15.13 Suppose that both end points \( a \) and \( b \) are regular and \( \alpha = \) \( \left( {{\alpha }_{1},{\alpha }_{2}}\right) \), where \( {\alpha }_{1},{\alpha }_{2} \in \lbrack 0,\pi ) \) . Let \( {A}_{\alpha } \) denote the restriction of \( {T}^{ * } \) to the domain\n\n\[ \n\mathcal{D}\left( {A}... | Proof The operator \( {A}_{\alpha } \) is self-adjoint, because it is defined by Eqs. (14.34) in Example 14.8 applied to the boundary triplet (15.15) of Example 15.3.\n\nSuppose that \( z \in \rho \left( {A}_{\alpha }\right) \) . Then \( {u}_{a} \) and \( {u}_{b} \) are linearly independent, and hence the Wronskian is ... | Yes |
Proposition 15.14 Suppose that \( T \) is in the limit circle case at a and \( b \) . Assume that \( {u}_{1} \) and \( {u}_{2} \) are as in Example 15.4 (that is, \( {u}_{1} \) and \( {u}_{2} \) are real-valued functions of \( \mathcal{D}\left( {T}^{ * }\right) \) satisfying (15.19)). Let \( {\alpha }_{1},{\alpha }_{2}... | Proof Since conditions (15.32) are just Eqs. (14.34) in Example 14.8 applied to the boundary triplet (15.22) of Example 15.4, the operator \( {A}_{\alpha } \) is self-adjoint.\n\nFix \( z \in \rho \left( {A}_{\alpha }\right) \) . Let \( \left\{ {{v}_{1},{v}_{2}}\right\} \) be a fundamental system of the equation \( \ma... | Yes |
Corollary 15.15 Suppose that \( T \) is in the limit circle case at a and b. Then each self-adjoint extension \( A \) of \( T \) on \( {L}^{2}\left( {a, b}\right) \) has a purely discrete spectrum. If \( {v}_{n}\left( A\right) \) , \( n \in \mathbb{N} \), is an enumeration of all nonzero eigenvalues of \( A \) counted ... | Proof Recall that the resolvent \( {R}_{\mathrm{i}}\left( {A}_{\alpha }\right) \) of the operator \( {A}_{\alpha } \) from Proposition 15.14 is an integral operator with kernel \( {K}_{\mathrm{i}} \in {L}^{2}\left( {\left( {a, b}\right) \times \left( {a, b}\right) }\right) \), since \( {u}_{a},{u}_{b} \in {L}^{2}\left(... | Yes |
Explicit examples of indeterminate moment sequences were already constructed by T. Stieltjes (1894). He observed that the measures \( \mu ,{\mu }_{k} \in \mathcal{M}\left( \mathbb{R}\right), k \in \mathbb{Z} \), given by\n\n\[ \n{d\mu }\left( x\right) = {\chi }_{\left( 0, + \infty \right) }\left( x\right) {e}^{-{x}^{1/... | Substituting \( t = \ln x \), the moments of \( {\mu }_{k}, k \in \mathbb{Z} \), are computed by\n\n\[ \n{s}_{n}\left( {\mu }_{k}\right) = {\int }_{\mathbb{R}}{x}^{n}d{\mu }_{k}\left( x\right) = {\int }_{0}^{\infty }{x}^{n}{x}^{k - \ln x}{dx} = {\int }_{\mathbb{R}}{e}^{tn}{e}^{t\left( {k - t}\right) }{e}^{t}{dt} \n\]\n... | Yes |
Theorem 16.1 Let \( s \) be a positive definite real sequence. Then the moment problem for \( s \) has a solution. | Proof Let \( A \) be a self-adjoint extension of \( {M}_{x} \) on \( \mathcal{G} \) . Since \( {M}_{x} \subseteq A \) and hence \( {\left( {M}_{x}\right) }^{n} \subseteq {A}^{n} \), the polynomial 1 is in the domain \( \mathcal{D}\left( {A}^{n}\right) \), and we have\n\n\[ \n{\int }_{\mathbb{R}}{x}^{n}d\left\langle {{E... | Yes |
Lemma 16.2 Let \( \mu \in {\mathcal{M}}_{s} \) . Then we have\n\n\[ \n{L}_{s}\left( p\right) = {\int }_{\mathbb{R}}p\left( x\right) {d\mu }\left( x\right) \;\text{ and }\;\langle p, q{\rangle }_{s} = \langle p, q{\rangle }_{\mu }\;\text{ for }p, q \in \mathbb{C}\left\lbrack x\right\rbrack .\n\] | Proof From \( \mu \in \mathcal{M}\left( \mathbb{R}\right) \) it follows that \( \mathbb{C}\left\lbrack x\right\rbrack \subseteq \mathcal{D}\left( {A}_{x}\right) \) . Clearly, \( {M}_{x} \subseteq {A}_{x} \) . Since \( \mu \) is a representing measure for \( s \), we have\n\n\[ \n{L}_{s}\left( p\right) = \mathop{\sum }\... | Yes |
Proposition 16.3 There exists an orthonormal basis \( \\left\\{ {{P}_{n}\\left( x\\right) : n \\in {\\mathbb{N}}_{0}}\\right\\} \) of the unitary space \( \\left( {\\mathbb{C}\\left\\lbrack x\\right\\rbrack ,\\langle \\cdot , \\cdot {\\rangle }_{s}}\\right) \) such that degree \( {P}_{n} = n \) and the leading coeffici... | Proof For the existence, it suffices to apply the Gram-Schmidt procedure to the vector space basis \( \\left\\{ {1, x,{x}^{2},\\ldots }\\right\\} \) of the unitary space \( \\left( {\\mathbb{C}\\left\\lbrack x\\right\\rbrack ,\\langle \\cdot , \\cdot {\\rangle }_{s}}\\right) \) . Since the scalar product is real for po... | Yes |
Proposition 16.4 For \( n \in \mathbb{N} \), the polynomial \( {P}_{n}\left( x\right) \) has \( n \) distinct real zeros. | Proof Let \( n \in {\mathbb{N}}_{0} \) . Let \( {\lambda }_{1},\ldots ,{\lambda }_{k} \) be the real points, where \( {P}_{n} \) changes sign, and put \( r\left( x\right) \mathrel{\text{:=}} \left( {x - {\lambda }_{1}}\right) \ldots \left( {x - {\lambda }_{k}}\right) \) . If there is no such point, we set \( r = 1 \) .... | Yes |
Proposition 16.5 There are numbers \( {a}_{n} > 0 \) and \( {b}_{n} \in \mathbb{R} \) for \( n \in {\mathbb{N}}_{0} \) such that\n\n\[ \n{M}_{x}{P}_{n} \equiv x{P}_{n}\left( x\right) = {a}_{n}{P}_{n + 1}\left( x\right) + {b}_{n}{P}_{n}\left( x\right) + {a}_{n - 1}{P}_{n - 1}\left( x\right) ,\;n \in {\mathbb{N}}_{0},\le... | Proof Let \( n \in {\mathbb{N}}_{0} \) . By Proposition 16.3, \( x{P}_{n}\left( x\right) \) has degree \( n + 1 \), and \( \left\{ {{P}_{0},\ldots ,{P}_{n + 1}}\right\} \) is a basis of the vector space of real polynomials of degree less than or equal to \( n + 1 \) . Hence, there are real numbers \( {c}_{nk} \) such t... | Yes |
Lemma 16.6 Let \( \gamma = \left( {\gamma }_{n}\right) ,\beta = \left( {\beta }_{n}\right) \) be complex sequences, and \( z, w \in \mathbb{C} \) . Then\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}\left\lbrack {{\left( \mathcal{T}\gamma \right) }_{k}{\beta }_{k} - {\gamma }_{k}{\left( \mathcal{T}\beta \right) }_{k})}\righ... | Proof We prove the first identity (16.11) by computing\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}\left\lbrack {{\left( \mathcal{T}\gamma \right) }_{k}{\beta }_{k} - {\gamma }_{k}{\left( \mathcal{T}\beta \right) }_{k})}\right\rbrack \]\n\n\[ = \left( {{a}_{0}{\gamma }_{1} + {b}_{0}{\gamma }_{0}}\right) {\beta }_{0} - {\g... | Yes |
Proposition 16.7 The adjoint operator \( {T}^{ * } \) is given by\n\n\[ \n{T}^{ * }\gamma = \mathcal{T}\gamma \;\text{ for }\gamma \in \mathcal{D}\left( {T}^{ * }\right) = \left\{ {\gamma \in {l}^{2}\left( {\mathbb{N}}_{0}\right) : \mathcal{T}\gamma \in {l}^{2}\left( {\mathbb{N}}_{0}\right) }\right\} .\n\]\n\nFor \( \g... | Proof Let \( \gamma \in {l}^{2}\left( {\mathbb{N}}_{0}\right) \) be such that \( \mathcal{T}\gamma \in {l}^{2}\left( {\mathbb{N}}_{0}\right) \) . A straightforward computation shows that \( \langle {T\beta },\gamma \rangle = \langle \beta ,\mathcal{T}\gamma \rangle \) for \( \beta \in \mathrm{d} \) . Therefore, \( \gam... | Yes |
Lemma 16.8 Let \( z \in \mathbb{C} \) . (i) \( \mathcal{N}\left( {{T}^{ * } - {zI}}\right) = \{ 0\} \) if \( {\mathfrak{p}}_{z} \notin {l}^{2}\left( {\mathbb{N}}_{0}\right) \) and \( \mathcal{N}\left( {{T}^{ * } - {zI}}\right) = \mathbb{C} \cdot {\mathfrak{p}}_{z} \) if \( {\mathfrak{p}}_{z} \in {l}^{2}\left( {\mathbb{... | Proof (i): From Proposition 16.7 it follows that a sequence \( \gamma \) is in \( \mathcal{N}\left( {{T}^{ * } - {zI}}\right) \) if and only if \( \gamma \in {l}^{2}\left( {\mathbb{N}}_{0}\right) ,\mathcal{T}\gamma \in {l}^{2}\left( {\mathbb{N}}_{0}\right) \) and (16.14) holds for all \( n \in {\mathbb{N}}_{0} \), wher... | Yes |
Corollary 16.9 The symmetric operator \( T \) (or \( {M}_{x} \) ) has deficiency indices \( \left( {0,0}\right) \) or \( \left( {1,1}\right) \) . The operator \( T\left( {\text{or}{M}_{x}}\right) \) is essentially self-adjoint if and only if \( {\mathfrak{p}}_{z} \) is not in \( {l}^{2}\left( {\mathbb{N}}_{0}\right) \)... | Proof Since \( {P}_{n}\left( x\right) \in \mathbb{R}\left\lbrack x\right\rbrack \) and hence \( \overline{{P}_{n}\left( z\right) } = {P}_{n}\left( \bar{z}\right) \), we have \( {\mathfrak{p}}_{z} \in {l}^{2}\left( {\mathbb{N}}_{0}\right) \) if and only if \( {\mathfrak{p}}_{\bar{z}} \in {l}^{2}\left( {\mathbb{N}}_{0}\r... | Yes |
Lemma 16.10 If \( A \) and \( B \) are different self-adjoint extensions of the multiplication operator \( {M}_{x} \) on \( {\mathcal{H}}_{s} \), then \( {\left\langle {\left( A - zI\right) }^{-1}1,1\right\rangle }_{s} \neq {\left\langle {\left( B - zI\right) }^{-1}1,1\right\rangle }_{s} \) for all \( z \in \mathbb{C} ... | Proof Fix \( z \in \mathbb{C} \smallsetminus \mathbb{R} \) and assume to the contrary that\n\n\[ \n{\left\langle {\left( A - zI\right) }^{-1}1,1\right\rangle }_{s} = {\left\langle {\left( B - zI\right) }^{-1}1,1\right\rangle }_{s}.\n\]\n\n(16.16)\n\nPut \( f \mathrel{\text{:=}} {\left( A - zI\right) }^{-1}1 - {\left( B... | Yes |
Proposition 16.12 If there exist a number \( {z}_{0} \in \mathbb{C} \smallsetminus \mathbb{R} \) and a sequence \( {\left( {r}_{n}\left( x\right) \right) }_{n \in \mathbb{N}} \) of polynomials \( {r}_{n} \in \mathbb{C}\left\lbrack x\right\rbrack \) such that\n\n\[1 = \mathop{\lim }\limits_{{n \rightarrow \infty }}\left... | Proof Fix \( p \in \mathbb{C}\left\lbrack x\right\rbrack \) . Since \( {z}_{0} \) is a zero of the polynomial \( p\left( x\right) - p\left( {z}_{0}\right) \), there is a polynomial \( q \in \mathbb{C}\left\lbrack x\right\rbrack \) such that \( p\left( x\right) - p\left( {z}_{0}\right) = \left( {x - {z}_{0}}\right) q\le... | Yes |
Lemma 16.14 \( \mathcal{T}{\mathfrak{q}}_{z} = {e}_{0} + z{\mathfrak{q}}_{z} \) for all \( z \in \mathbb{C} \) | Proof By the recurrence relation for \( {Q}_{n}\left( z\right) ,{\left( \mathcal{T}{\mathfrak{q}}_{z}\right) }_{n} = z{Q}_{n}\left( z\right) \equiv z{\left( {\mathfrak{q}}_{z}\right) }_{n} \) for \( n \in \mathbb{N} \) . Using that \( {\gamma }_{0} = 0 \) and \( {\gamma }_{1} = {a}_{0}^{-1} \), we compute the zero comp... | Yes |
Corollary 16.16 Let \( n \in \mathbb{N} \) and \( z,{z}^{\prime } \in \mathbb{C} \) . If \( \left| {\operatorname{Im}z}\right| \leq \left| {\operatorname{Im}{z}^{\prime }}\right| \), then\n\n\[ \left| {{P}_{n}\left( z\right) }\right| \leq \left| {{P}_{n}\left( {z}^{\prime }\right) }\right| \;\text{ and }\;\left| {{Q}_{... | Proof Let \( r \) be a polynomial of degree \( n \) which has \( n \) distinct real zeros, say \( {x}_{1},\ldots ,{x}_{n} \) . Then \( r\left( z\right) = c\left( {z - {x}_{1}}\right) \ldots \left( {z - {x}_{n}}\right) \) for some \( c \in \mathbb{C} \) . Hence, it is easily seen that \( \left| {r\left( z\right) }\right... | Yes |
Proposition 16.17 \( {Q}_{n}\left( z\right) = {L}_{s, x}\left( \frac{{P}_{n}\left( x\right) - {P}_{n}\left( z\right) }{x - z}\right) \) for \( n \in {\mathbb{N}}_{0} \) and \( z \in \mathbb{C} \) . | Proof Let us denote the polynomial at the right-hand side by \( {r}_{n}\left( z\right) \) . From the recurrence relation (16.5) we obtain for \( n \in \mathbb{N} \) ,\n\n\[ \n{a}_{n}\frac{{P}_{n + 1}\left( x\right) - {P}_{n + 1}\left( z\right) }{x - z} + {b}_{n}\frac{{P}_{n}\left( x\right) - {P}_{n}\left( z\right) }{x ... | Yes |
Proposition 16.18 Let \( \mu \in {\mathcal{M}}_{s} \) . For \( z \in \mathbb{C} \smallsetminus \mathbb{R} \) and \( n \in {\mathbb{N}}_{0} \) ,\n\n\[ \n{\left\langle {f}_{z},{P}_{n}\right\rangle }_{{L}^{2}\left( {\mathbb{R},\mu }\right) } = {Q}_{n}\left( z\right) + {I}_{\mu }\left( z\right) {P}_{n}\left( z\right) ,\n\]... | Proof Clearly, the bounded function \( {f}_{z}\left( x\right) = \frac{1}{x - z} \) is in \( {L}^{2}\left( {\mathbb{R},\mu }\right) \) . We compute\n\n\[ \n{\left\langle {f}_{z},{P}_{n}\right\rangle }_{{L}^{2}\left( {\mathbb{R},\mu }\right) } = {\int }_{\mathbb{R}}\frac{{P}_{n}\left( x\right) - {P}_{n}\left( z\right) }{... | Yes |
Lemma 16.19 For \( z, w \in \mathbb{C} \) and \( k \in {\mathbb{N}}_{0} \), we have\n\n\[ \n{A}_{k}\left( {z, w}\right) \mathrel{\text{:=}} \left( {z - w}\right) \mathop{\sum }\limits_{{n = 0}}^{k}{Q}_{n}\left( z\right) {Q}_{n}\left( w\right) = {a}_{k}\left( {{Q}_{k + 1}\left( z\right) {Q}_{k}\left( w\right) - {Q}_{k}\... | Proof All four identities are easily derived from Eq. (16.12). As a sample, we verify the identity for \( {B}_{k}\left( {z, w}\right) \) . Recall that the sequences \( {\mathfrak{p}}_{z} = \left( {{P}_{n}\left( z\right) }\right) \) and \( {\mathfrak{q}}_{w} = \left( {{Q}_{n}\left( w\right) }\right) \) satisfy the relat... | Yes |
For any \( z, w \in \mathbb{C} \), we have\n\n\[ \n{A}_{k}\left( {z, w}\right) {D}_{k}\left( {z, w}\right) - {B}_{k}\left( {z, w}\right) {C}_{k}\left( {z, w}\right) = 1, \n\] | Proof Inserting the four identities from Lemma 16.19, we compute\n\n\[ \n{A}_{k}\left( {z, w}\right) {D}_{k}\left( {z, w}\right) - {B}_{k}\left( {z, w}\right) {C}_{k}\left( {z, w}\right) \n\]\n\n\[ \n= {a}_{k}^{2}\left( {{P}_{k + 1}\left( z\right) {Q}_{k}\left( z\right) - {P}_{k}\left( z\right) {Q}_{k + 1}\left( z\righ... | Yes |
Corollary 16.22 Ifs is an indeterminate moment sequence, then for all (!) numbers \( z \in \mathbb{C} \), the sequences \( {\mathfrak{p}}_{z} = {\left( {P}_{n}\left( z\right) \right) }_{n \in {\mathbb{N}}_{0}} \) and \( {\mathfrak{q}}_{z} = {\left( {Q}_{n}\left( z\right) \right) }_{n \in {\mathbb{N}}_{0}} \) are in \( ... | \[ {T}^{ * }{\mathfrak{p}}_{z} = z{\mathfrak{p}}_{z}\;\text{ and }\;{T}^{ * }{\mathfrak{q}}_{z} = {e}_{0} + z{\mathfrak{q}}_{z}. \] | Yes |
For any sequence \( c = \left( {c}_{n}\right) \in {l}^{2}\left( {\mathbb{N}}_{0}\right) \), the equations\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{P}_{n}\left( z\right) \;\text{ and }\;g\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{Q}_{n}\left( z\right) \]\n\n(16.26... | Proof We carry out the proof for \( f\left( z\right) \) . Since \( \left( {{P}_{n}\left( z\right) }\right) \in {l}^{2}\left( {\mathbb{N}}_{0}\right) \) by Lemma 16.21 and \( \left( {c}_{n}\right) \in {l}^{2}\left( {\mathbb{N}}_{0}\right) \), the series \( f\left( z\right) \) converges for all \( z \in \mathbb{C} \) . W... | Yes |
Proposition 16.24 Let \( z, w \in \mathbb{C} \) . Then we have:\n\n(i) \( A\left( {z, w}\right) D\left( {z, w}\right) - B\left( {z, w}\right) C\left( {z, w}\right) = 1 \) . | Proof (i) and (ii) follow from Lemma 16.20 by passing to the limit \( k \rightarrow \infty \), while (iii) is easily obtained from the definitions of \( A, B, C, D \) and \( {\mathfrak{p}}_{z},{\mathfrak{q}}_{w} \) . | No |
Proposition 16.25 If \( \mu \in {\mathcal{M}}_{s} \) is a von Neumann solution, then\n\n\[ \n{I}_{\mu }\left( z\right) = - \frac{A\left( {z, w}\right) + {I}_{\mu }\left( w\right) C\left( {z, w}\right) }{B\left( {z, w}\right) + {I}_{\mu }\left( w\right) D\left( {z, w}\right) }\;\text{ for }z, w \in \mathbb{C} \smallsetm... | Proof Since \( \mu \) is a von Neumann solution, \( {\mathcal{H}}_{s} \cong {L}^{2}\left( {\mathbb{R},\mu }\right) \) . Hence, \( \left\{ {{P}_{n} : n \in {\mathbb{N}}_{0}}\right\} \) is an orthonormal basis of \( {L}^{2}\left( {\mathbb{R},\mu }\right) \), so by (16.21) for all \( z, w \in \mathbb{C} \smallsetminus \ma... | Yes |
Corollary 16.27 The self-adjoint extensions of the operator \( T \) on the Hilbert space \( {\mathcal{H}}_{s} \cong {l}^{2}\left( {\mathbb{N}}_{0}\right) \) are the operators \( {T}_{\left( t\right) } = {T}^{ * } \mid \mathcal{D}\left( {T}_{\left( t\right) }\right), t \in \mathbb{R} \cup \{ \infty \} \), where\n\n\[ \m... | Proof By Example 14.7 the self-adjoint extensions of \( T \) on \( {l}^{2}\left( {\mathbb{N}}_{0}\right) \) are the operators \( {T}_{t}, t \in \mathbb{R} \cup \{ \infty \} \), where \( \mathcal{D}\left( {T}_{t}\right) = \left\{ {\varphi \in \mathcal{D}\left( {T}^{ * }\right) : t{\Gamma }_{0}\varphi = {\Gamma }_{1}\var... | Yes |
Lemma 16.29 \( \mathop{\lim }\limits_{{y \in \mathbb{R}, y \rightarrow 0}}{I}_{{\mu }_{t}}\left( {y\mathrm{i}}\right) = \mathop{\lim }\limits_{{y \in \mathbb{R}, y \rightarrow 0}}\left\langle {{\left( {T}_{\left( t\right) } - y\mathrm{i}I\right) }^{-1}{e}_{0},{e}_{0}}\right\rangle = t \) for \( t \in \mathbb{R} \) . | Proof Let \( t \in \mathbb{R} \) . Since \( \mathcal{N}\left( {T}_{\left( t\right) }\right) \subseteq \mathcal{N}\left( {T}^{ * }\right) = \mathbb{C} \cdot {\mathfrak{p}}_{0} \) by Lemma 16.8(i) and \( {\mathfrak{p}}_{0} \notin \) \( \mathcal{D}\left( {T}_{\left( t\right) }\right) \) by (16.31) and (16.29), we have \( ... | Yes |
Corollary 16.30 Let \( t \in \mathbb{R} \cup \{ \infty \} \) . Then \( {I}_{{\mu }_{t}}\left( z\right) \) is a meromorphic function, and the self-adjoint operator \( {T}_{\left( t\right) } \) has a purely discrete spectrum consisting of eigenvalues of multiplicity one. The eigenvalues of \( {T}_{\left( t\right) } \) ar... | Proof Since \( A\left( z\right), B\left( z\right), C\left( z\right), D\left( z\right) \) are entire functions,(16.32) implies that \( {I}_{{\mu }_{t}}\left( z\right) \) is meromorphic. By Theorem F. 5 the support of \( {\mu }_{t} \) is the set of poles of the meromorphic function \( {I}_{{\mu }_{t}} \) . Therefore, sin... | Yes |
A map \( f : X \rightarrow Y \) between two topological spaces is continuous at \( \bar{x} \in X \) if and only if for every net \( {\left( {x}_{i}\right) }_{i \in I} \) of \( X \) converging to \( \bar{x} \), the net \( {\left( f\left( {x}_{i}\right) \right) }_{i \in I} \) converges to \( f\left( \bar{x}\right) \) . S... | Necessity is immediate. Let us show sufficiency. Suppose \( f \) is not continuous at \( \bar{x} \) . Then there exists \( V \in \mathcal{N}\left( {f\left( \bar{x}\right) }\right) \) such that for all \( U \in \mathcal{N}\left( \bar{x}\right) \) there exists some \( {x}_{U} \in U \) with \( f\left( {x}_{U}\right) \noti... | Yes |
Proposition 1.4. The set of continuous linear forms on \( {X}^{ * } \) endowed with the weak* topology can be identified with \( X \) . | Proof. By definition, for all \( x \in X \), the linear form \( {e}_{x} : {x}^{ * } \mapsto \left\langle {{x}^{ * }, x}\right\rangle \) on \( {X}^{ * } \) is continuous for the weak* topology. Let us show that every continuous linear form \( f \) on \( \left( {{X}^{ * },{w}^{ * }}\right) \) coincides with some \( {e}_{... | Yes |
Theorem 1.5 (Alaoglu-Bourbaki). Every weak* closed, bounded subset of the dual space \( {X}^{ * } \) is weak* compact, i.e., is compact for the weak* topology. | Proof. It suffices to show that the closed unit ball \( {B}^{ * } \mathrel{\text{:=}} {B}_{{X}^{ * }} \) of \( {X}^{ * } \) is weak* compact. To do so, let us denote by \( S \) the closed unit sphere \( {S}_{X} \) of \( X \), by \( H \) the space of positively homogeneous functions on \( X \), and by \( {H}_{S} \) the ... | Yes |
Theorem 1.13. The bounded weak* topology \( {\mathcal{W}}_{b} \) on the dual \( {X}^{ * } \) of a Banach space \( X \) coincides with the topology of uniform convergence on compact subsets of \( X \) . | Proof. First, let us prove that for every compact subset \( K \) of \( X \) the set \( {K}^{0} \) is a neighborhood of 0 for the topology \( {\mathcal{W}}_{b} \) . It suffices to show that\n\n\[ V \mathrel{\text{:=}} \left\{ {{x}^{ * } \in {X}^{ * } : \forall x \in K{x}^{ * }\left( x\right) < 1}\right\} \]\n\nis open f... | Yes |
Proposition 1.15. For a function \( f : X \rightarrow \overline{\mathbb{R}} \), the following assertions are equivalent:\n\n(a) \( f \) is lower semicontinuous;\n\n(b) The epigraph \( E \mathrel{\text{:=}} {E}_{f} \mathrel{\text{:=}} \{ \left( {x, r}\right) \in X \times \mathbb{R} : r \geq f\left( x\right) \} \) of \( ... | Proof. (a) \( \Rightarrow \) (b) It suffices to prove that \( \left( {X \times \mathbb{R}}\right) \smallsetminus E \) is open when \( f \) is lower semicontinuous. Given \( \left( {\bar{x},\bar{r}}\right) \in \left( {X \times \mathbb{R}}\right) \smallsetminus E \), i.e., such that \( \bar{r} < f\left( \bar{x}\right) \)... | Yes |
For \( f : X \rightarrow \overline{\mathbb{R}}, r \in {\mathbb{R}}_{ + } \), one has \( \mathop{\liminf }\limits_{{x \rightarrow w}}{rf}\left( x\right) = r\mathop{\liminf }\limits_{{x \rightarrow w}}f\left( x\right) \) . If \( f, g : X \rightarrow \overline{\mathbb{R}} \) are such that \( \left\{ {\mathop{\liminf }\lim... | The first assertion being immediate, let us establish the second one. Let us set \( \bar{f}\left( w\right) \mathrel{\text{:=}} \mathop{\liminf }\limits_{{x \rightarrow w}}f\left( x\right) \) and \( \bar{g}\left( w\right) \mathrel{\text{:=}} \mathop{\liminf }\limits_{{x \rightarrow w}}g\left( x\right) \) . If \( \bar{f}... | Yes |
Lemma 1.18. A function \( f : X \rightarrow \overline{\mathbb{R}} \) on a topological space \( X \) is lower semicontinuous at some \( w \in X \) if and only if one has \( f\left( w\right) \leq \mathop{\liminf }\limits_{{x \rightarrow w}}f\left( x\right) \) . | Proof. Here \( W = X \) . Clearly, when \( f \) is lower semicontinuous at \( w \), one has \( f\left( w\right) \leq \) \( \mathop{\liminf }\limits_{{x \rightarrow w}}f\left( x\right) \) . Conversely, when this inequality holds, for every \( r < f\left( w\right) \), by the definition of the supremum over \( \mathcal{N}... | Yes |
A function \( f : X \rightarrow \overline{\mathbb{R}} \) on a topological space \( X \) is lower semicontinuous at some \( \bar{x} \in X \) if and only if for every net \( {\left( {x}_{i}\right) }_{i \in I} \) in \( X \) converging to \( \bar{x} \) one has \( f\left( \bar{x}\right) \leq \mathop{\liminf }\limits_{{i \in... | Proof. The condition is necessary: if \( f \) is lower semicontinuous at \( \bar{x} \) and if a net \( {\left( {x}_{i}\right) }_{i \in I} \) in \( X \) converges to \( \bar{x} \), then for all \( r < f\left( \bar{x}\right) \) there exists some \( V \in \mathcal{N}\left( \bar{x}\right) \) such that \( f\left( v\right) >... | Yes |
Proposition 1.20. If \( {\left( {f}_{i}\right) }_{i \in I} \) is a family of functions that are lower semicontinuous at \( \bar{x} \), then the function \( f \mathrel{\text{:=}} \mathop{\sup }\limits_{{i \in I}}{f}_{i} \) is lower semicontinuous at \( \bar{x} \). | Proof. Let \( r \in \mathbb{R}, r < f\left( \bar{x}\right) \) . There exists some \( j \in I \) such that \( r < {f}_{j}\left( \bar{x}\right) \) ; hence one can find some \( V \in \mathcal{N}\left( \bar{x}\right) \) such that \( r < {f}_{j}\left( v\right) \leq f\left( v\right) \) for all \( v \in V \) . The proofs of t... | No |
For every function \( f : X \rightarrow \overline{\mathbb{R}} \) on a topological space \( X \), the family of lower semicontinuous functions majorized by \( f \) has a greatest element \( \bar{f} \) called the lower semicontinuous hull of \( f \). Its epigraph is the closure of the epigraph of \( f \). The function \(... | The first assertion is a direct consequence of Proposition 1.20. The second one easily stems from the fact that the closure of the epigraph of \( f \) is the epigraph of a function. The proof of the explicit expression of \( \bar{f} \) is left as an exercise. | No |
Proposition 1.22. Let \( W, X \) be topological spaces, let \( \bar{w} \in W \), and let \( f : W \times \) \( X \rightarrow \overline{\mathbb{R}} \) be a function that is lower semicontinuous at \( \left( {\bar{w},\bar{x}}\right) \) for every \( \bar{x} \in X \) . If the following compactness assumption is satisfied, ... | Proof. Given a net \( {\left( {w}_{i}\right) }_{i \in I} \rightarrow \bar{w} \), one can find a subnet \( {\left( {w}_{j}\right) }_{j \in J} \) such that \( {\left( p\left( {w}_{j}\right) \right) }_{j \in J} \) converges to \( \mathop{\liminf }\limits_{{i \in I}}p\left( {w}_{i}\right) \) and (taking a further subnet if... | Yes |
Corollary 1.23. Let \( W \) and \( X \) be topological spaces, \( X \) being compact, and let \( f \) : \( W \times X \rightarrow {\mathbb{R}}_{\infty } \mathrel{\text{:=}} \mathbb{R} \cup \{ + \infty \} \) be lower semicontinuous at all points of \( \{ \bar{w}\} \times X \) . Then the performance function \( p \) defi... | Proof. Condition (C) is clearly satisfied when \( X \) is compact, since for every net \( {\left( {w}_{i}\right) }_{i \in I} \rightarrow \bar{w} \) and for every sequence \( \left( {\alpha }_{n}\right) \rightarrow {0}_{ + } \) one can take \( H \mathrel{\text{:=}} I \times \mathbb{N},{w}_{h} \mathrel{\text{:=}} {w}_{i}... | Yes |
Theorem 1.24 (Weierstrass). Let \( f : X \rightarrow \overline{\mathbb{R}} \) be a lower semicontinuous function on a compact topological space \( X \) . Then the set \( M \mathrel{\text{:=}} \{ w \in X : f\left( w\right) \leq f\left( x\right) \forall x \in X\} \) of minimizers of \( f \) is nonempty. | Proof. We may suppose \( m \mathrel{\text{:=}} \inf f\left( X\right) < + \infty \), for otherwise, \( f \) is constant with value \( + \infty \) . Setting \( {S}_{f}\left( r\right) \mathrel{\text{:=}} \{ x \in X : f\left( x\right) \leq r\} \), the family \( \left\{ {{S}_{f}\left( r\right) : r > m}\right\} \) is formed ... | Yes |
Corollary 1.25. Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be an inf-compact function on a topological space X. Then \( f \) attains its minimum. | Proof. The result being obvious when \( f \) takes the value \( + \infty \) only, let \( r \in \mathbb{R} \) be such that \( {S}_{f}\left( r\right) \mathrel{\text{:=}} {f}^{-1}(\left( {-\infty, r\rbrack }\right) \) is nonempty. By assumption, \( {S}_{f}\left( r\right) \) is compact. By Theorem 1.24, \( f \) attains its... | Yes |
Lemma 1.26 (Baire's theorem). In a complete metric space \( T \), every generic subset is dense. Moreover, if \( T \) is the union of a countable family of closed subsets, then one of them has a nonempty interior. | Proof. Let us show that every subset \( G \) of \( T \) containing the intersection \( \mathop{\bigcap }\limits_{n}{T}_{n} \) of a countable family of open dense subsets \( {T}_{n} \) of \( T \) is dense. Let \( \left( {s}_{n}\right) \) be a sequence of positive numbers with limit 0 . Given a nonempty open subset \( U ... | Yes |
Theorem 1.27 (Banach–Steinhaus or uniform boundedness theorem). Let \( X \) , \( Y \) be normed spaces, \( X \) being complete, and let \( F \) be a subset of the space \( L\left( {X, Y}\right) \) of continuous linear maps from \( X \) to \( Y \) . If for all \( x \in X \), the set \( F\left( x\right) \mathrel{\text{:=... | Proof. Denoting by \( {B}_{Y} \) unit ball of \( Y \), for \( n \in \mathbb{N} \) define the closed set\n\n\[ \n{X}_{n} \mathrel{\text{:=}} \{ x \in X : \forall f \in F\parallel f\left( x\right) \parallel \leq n\} = \mathop{\bigcap }\limits_{{f \in F}}{f}^{-1}\left( {n{B}_{Y}}\right) .\n\]\n\nBy assumption, \( X \) is ... | Yes |
Corollary 1.28. A weak* bounded subset of the dual \( {X}^{ * } \) of a Banach space \( X \) is bounded. A weakly bounded subset of \( X \) is bounded. In particular, a weak* convergent sequence of \( {X}^{ * } \) is bounded. | Proof. The first assertion is the special case of the theorem corresponding to \( Y \mathrel{\text{:=}} \mathbb{R} \) . The second one stems from the fact that the embedding of \( X \) into \( {X}^{* * } \) is isometric, as a consequence of the Hahn-Banach theorem, which we will prove below. | No |
Proposition 1.30. [26] Let \( X \) and \( Y \) be two linear spaces (or additive groups), let \( c, d \) in \( Y \), and let \( A, B : X \rightrightarrows Y \) be two multimaps. Then the equations\n\n\[ c + d \in A\\left( x\\right) + B\\left( x\\right) \]\n\n(1.2)\n\n\[ 0 \in {A}^{-1}\\left( {y + c}\\right) - {B}^{-1}\... | Proof. Clearly, \( x \) is a solution to (1.2) iff there exists \( y \in A\\left( x\\right) - c \) such that \( d - y \in \) \( B\\left( x\\right) \) . This amounts to saying that \( x \in {A}^{-1}\\left( {y + c}\\right) \cap {B}^{-1}\\left( {d - y}\\right) \) or that (1.3) holds. In other words, \( y + c \in A\\left( ... | Yes |
Proposition 1.35. (a) If \( X \) is a regular (resp. Hausdorff) space, if \( F : T \rightrightarrows X \) is outward continuous at some point \( {t}_{0} \in T \), and if \( F\left( {t}_{0}\right) \) is closed (resp. compact), then \( F \) is closed at \( {t}_{0} \) . | Proof. (a) When \( F\left( {t}_{0}\right) \) is closed and \( X \) is regular, given \( x \in X \smallsetminus F\left( {t}_{0}\right) \) there exist neighborhoods \( V \) of \( x, W \) of \( F\left( {t}_{0}\right) \) that are disjoint (take for \( V \) a closed neighborhood of \( x \) contained in \( X \smallsetminus F... | No |
Corollary 1.36. If \( F : T \rightrightarrows X \) is closed at \( {t}_{0} \), then \( F \) is compactly outward continuous at \( {t}_{0} \) and \( F\left( {t}_{0}\right) \) is closed. The converse holds when \( {t}_{0} \) and the points of \( X \) have a countable base of neighborhoods. | Proof. If \( F : T \rightrightarrows X \) is closed at \( {t}_{0} \), then \( F\left( {t}_{0}\right) \) is closed, and for every compact subset \( K \) of \( X \) the multimap \( {F}_{K} : t \rightrightarrows F\left( t\right) \cap K \) is closed at \( {t}_{0} \), hence is outward continuous at \( {t}_{0} \) by Proposit... | Yes |
Theorem 1.40 (Michael [703]). Let \( T \) be a topological space that is either metrizable or compact and let \( F : T \rightrightarrows X \) be an inward continuous multimap with closed convex images in a Banach space \( X \) . Then \( F \) admits a continuous selection, i.e., a continuous map \( f : T \rightarrow X \... | Since the construction of \( f \) uses a partition of unity on \( T \), it is valid in fact when \( T \) belongs to the class of paracompact spaces, a class encompassing both metrizable and compact spaces. | No |
Proposition 1.43. If there exist \( S \in \mathcal{T} \mathrel{\text{:=}} \left\{ {U \cap T : U \in {\mathcal{N}}_{P}\left( 0\right) }\right\} \) and a map \( f : S \rightarrow X \) such that \( \mathop{\lim }\limits_{{s\left( { \in S}\right) \rightarrow 0}}f\left( s\right) = x \) and \( f\left( s\right) \in F\left( s\... | Proof. The first assertion follows from the definitions.\n\nConversely, assume that \( X \) is metrizable and let \( d \) be a compatible metric. Let \( x \in \mathop{\liminf }\limits_{{t\left( { \in T}\right) \rightarrow 0}}F\left( t\right) \) . If each \( F\left( t\right) \) is nonempty (that occurs for \( t \) close... | Yes |
Corollary 1.45. Given a multimap \( F : X \rightrightarrows Y \) between two topological spaces \( X \) and \( Y \) and \( \bar{x} \in X \) one has \( \mathop{\limsup }\limits_{{x \rightarrow \bar{x}}}F\left( x\right) \subset F\left( \bar{x}\right) \) if and only if \( F \) is closed at \( \bar{x} \) . | Since one always has \( F\left( \bar{x}\right) \subset \mathop{\limsup }\limits_{{x \rightarrow \bar{x}}}F\left( x\right) \), the first assertion can be rephrased as follows: \( F \) is closed at \( \bar{x} \) if and only if \( \lim \mathop{\sup }\limits_{{x \rightarrow \bar{x}}}F\left( x\right) = F\left( \bar{x}\right... | No |
Proposition 1.46. Let \( {\left( F\left( t\right) \right) }_{t \in T} \) be a parameterized family of subsets of \( X \) and let \( {F}_{0} \subset \mathop{\liminf }\limits_{{t\left( { \in T}\right) \rightarrow 0}}F\left( t\right) \) . Given a lower semicontinuous function \( h : X \rightarrow \overline{\mathbb{R}} \),... | Proof. The result is obvious when \( {m}_{0} = - \infty \) . Assume \( {m}_{0} > - \infty \) and take \( r \in \mathbb{R} \) , \( r < {m}_{0} \) . Since \( h \) is lower semicontinuous, the set \( V \mathrel{\text{:=}} {h}^{-1}(\left( {r, + \infty \rbrack }\right) \) is open in \( X \) and it meets \( {F}_{0} \) . Sinc... | Yes |
Corollary 1.47. Let \( F : T \rightrightarrows X \) be a parameterized family of sets as above and let \( {F}_{0} \subset \mathop{\liminf }\limits_{{t\left( { \in T}\right) \rightarrow 0}}F\left( t\right) \) . Given an outward continuous function \( j : X \rightarrow \overline{\mathbb{R}} \), let \( {p}_{0} \mathrel{\t... | Proof. It suffices to note that for \( h \mathrel{\text{:=}} - j \) one has \( p\left( t\right) = - m\left( t\right) ,{p}_{0} = - {m}_{0} \), where \( m \) is defined as in the preceding proposition. | Yes |
Proposition 1.48. Let \( F : T \rightrightarrows X \) be a parameterized family of sets as above, with \( T \subset P \) and \( 0 \in \operatorname{cl}\left( T\right) \), and let \( h : X \rightarrow \overline{\mathbb{R}} \) be outward continuous at some \( {x}_{0} \in {F}_{0} \subset X \) . Assume that \( {m}_{0} \leq... | Proof. Suppose, to the contrary, that \( h\left( {x}_{0}\right) < {m}_{0} \) . Let \( r,{r}^{\prime } \in \left( {h\left( {x}_{0}\right) ,{m}_{0}}\right) \) with \( r < \) \( {r}^{\prime } \) . Since \( h \) is outward continuous at \( {x}_{0} \) and \( {x}_{t} \rightarrow {x}_{0} \) as \( t \rightarrow 0 \) in \( T \)... | Yes |
Proposition 1.50. Let \( {f}_{0} \in {\overline{\mathbb{R}}}^{X},{\left( {f}_{t}\right) }_{t \in T} \) be such that \( e \) - \( \lim \mathop{\sup }\limits_{t}{f}_{t} \leq {f}_{0} \) . Then\n\n\[ \mathop{\limsup }\limits_{{t\left( { \in T}\right) \rightarrow 0}}\inf {f}_{t}\left( X\right) \leq \inf {f}_{0}\left( X\righ... | Proof. It suffices to apply Corollary 1.47 to the epigraph \( F\left( t\right) \) of \( {f}_{t} \) (resp. \( {F}_{0} \) of \( {f}_{0} \) ), taking for \( j \) the linear functional \( \left( {x, r}\right) \mapsto r \) . | Yes |
Lemma 1.51. Let \( W \) and \( X \) be linear spaces and let \( f : W \times X \rightarrow \overline{\mathbb{R}} \) be convex. Then the performance function \( p : W \rightarrow \overline{\mathbb{R}} \) defined as follows is convex:\n\n\[ p\left( w\right) \mathrel{\text{:=}} \mathop{\inf }\limits_{{x \in X}}f\left( {w,... | Proof. The result follows from the fact that the strict epigraph of \( p \) is the projection on \( W \times \mathbb{R} \) of the strict epigraph of \( f \) . | No |
Given a sequence \( {\left( {E}_{n}\right) }_{n \geq 1} \) of nonempty subsets of a linear space \( Z \), the convex hull \( C \) of the union \( E \) of the \( {E}_{n} \) ’s is the union over \( p \in \mathbb{N} \smallsetminus \{ 0\} \) of the convex hulls \( {C}_{p} \) of \( {E}_{1} \cup \cdots \cup {E}_{p} \) : | In fact, every element of \( C \) can be written as a convex combination of a finite family of elements of \( E \), hence is an element of \( {C}_{p} \) for some \( p \) . The reverse containment is obvious since \( {C}_{p} \subset C \) for all \( p \) .\n\nThe right-hand side of (1.10) is clearly contained in \( {C}_{... | Yes |
Lemma 1.57. If \( C \) is a nonempty convex subset of a finite-dimensional space, then \( C \) has a nonempty interior (called the relative interior and denoted by \( \operatorname{ri}\left( C\right) \) ) in the affine subspace \( A \) it generates. | Proof. By definition, \( A \) is the smallest affine subspace containing \( C \) . Using a translation, we may suppose \( 0 \in C \), so that \( A \) is the linear subspace generated by \( C \) . Let \( n \) be the dimension of \( A \) and let \( m \) be the greatest integer \( k \) such that there exists a linearly in... | Yes |
Lemma 1.58. For a nonempty convex subset \( C \) of a linear space \( X \) and \( u \in X \), the following assertions are equivalent:\n\n(a) \( u \in \operatorname{core}C \) ;\n\n(b) \( C - u \) is absorbing: for every \( x \in X \) there exists \( t > 0 \) such that \( {tx} \in C - u \) ;\n\n(c) \( X = {\mathbb{R}}_{... | Proof. The implications (a) \( \Rightarrow \) (b) \( \Rightarrow \) (c) are obvious. Now (c) implies that \( 0 \in C - u \) : this is obvious if \( X = \{ 0\} \), and otherwise, taking \( v \neq 0 \) in \( X \), we can write \( v = \) \( r\left( {c - u}\right) , - v = {r}^{\prime }\left( {{c}^{\prime } - u}\right) \), ... | Yes |
Proposition 1.59. The core of a convex subset \( C \) of a normed space \( X \) coincides with its interior int \( C \) whenever one of the following conditions is satisfied:\n\n(a) \( \operatorname{int}C \neq \varnothing \) ;\n\n(b) \( X \) is finite-dimensional;\n\n(c) \( X \) is a Banach space and \( C \) is closed. | Proof. The interior int \( C \) of a convex subset \( C \) of a normed space \( X \) is always contained in its core, since for every \( u \in \operatorname{int}C \) and every \( v \in X \) the map \( f : t \mapsto u + {tv} \) is continuous and \( f\left( 0\right) \in \operatorname{int}C \), so that \( f\left( t\right)... | Yes |
Lemma 1.61. Let \( W \) and \( X \) be Banach spaces, and let \( C \) be the projection \( {p}_{X}\left( F\right) \) on \( X \) of a closed convex subset \( F \) of \( W \times X \) . If the projection \( {p}_{W}\left( F\right) \) of \( F \) on \( W \) is bounded, then \( C \) is ideally convex. | Proof. Let \( x \) be the sum of a series with general term \( {t}_{n}{x}_{n} \), where \( \left( {t}_{n}\right) \in {\Delta }_{\infty },\left( {x}_{n}\right) \) is bounded and \( {x}_{n} \in C \) for all \( n \in \mathbb{N} \) . For all \( n \in \mathbb{N} \), there exists some \( {w}_{n} \in W \) such that \( \left( ... | Yes |
Lemma 1.62. Let \( C \) be an ideally convex subset of a Banach space \( X \) . Then\n\n\[ \operatorname{int}\left( C\right) = \operatorname{core}\left( C\right) = \operatorname{core}\left( {\operatorname{cl}\left( C\right) }\right) = \operatorname{int}\left( {\operatorname{cl}\left( C\right) }\right) . \] | Proof. By Proposition 1.59 (c), we already know that \( \operatorname{core}\left( {\operatorname{cl}\left( C\right) }\right) = \operatorname{int}\left( {\operatorname{cl}\left( C\right) }\right) \) . Given \( \bar{x} \in \operatorname{int}\left( {\operatorname{cl}\left( C\right) }\right) \), let us show that \( \bar{x}... | Yes |
Theorem 1.63 (Robinson-Ursescu). Let \( W, X \) be Banach spaces, let \( F : W \rightrightarrows X \) be a multimap with closed convex graph. Then for every \( \left( {\bar{w},\bar{x}}\right) \) in the graph of \( F \) such that \( \bar{x} \in \operatorname{core}F\left( W\right) \), the multimap \( F \) is open at \( \... | Proof. Without loss of generality, we may suppose \( \left( {\bar{w},\bar{x}}\right) = \left( {0,0}\right) \), so that \( F\left( W\right) \) is absorbing. Let \( B \) be the closed ball with center \( \bar{w} = 0 \) and radius \( r \) in \( W \) and let \( C \mathrel{\text{:=}} F\left( B\right) = {p}_{X}\left( {\left(... | Yes |
Theorem 1.64 (Finite-dimensional geometric form of the Hahn-Banach theorem). Let \( C \) be a nonempty convex subset of a finite-dimensional linear space \( X \) and let \( a \in X \smallsetminus C \) . Then there exists some \( f \in {X}^{ * } \smallsetminus \{ 0\} \) such that \( f\left( a\right) \geq \mathop{\sup }\... | Proof. (a) Let us first consider the case that \( C \) is closed. Since \( X \) is finite-dimensional, we may endow \( X \) with the norm associated with a scalar product \( \left( {\cdot \mid \cdot }\right) \) . Then the point \( a \) has a best approximation \( p \) in \( C \) characterized by \[ \forall z \in C,\;\l... | Yes |
Corollary 1.65. Let \( A \) and \( B \) be two disjoint nonempty convex subsets of a finite-dimensional space \( X \) . Then there exists some \( f \in {X}^{ * } \smallsetminus \{ 0\} \) such that \( f\left( a\right) \geq f\left( b\right) \) for all \( a \in A \) and all \( b \in B \) . | Proof. Since \( C \mathrel{\text{:=}} A - B \) is convex and since \( A \) and \( B \) are disjoint, one has \( 0 \notin C \) and it suffices to take the linear form \( f \) provided by the preceding statement. | Yes |
Proposition 1.66. The space \( S\left( X\right) \) of finitely valued sublinear functions on a linear space \( X \), ordered by the pointwise order, is lower inductive, hence has minimal elements. Each such element is a linear form. | Proof. We have to show that every totally ordered subset \( C \) of \( S\left( X\right) \) has a lower bound. Let \( {s}_{0} \) be a fixed element of \( C \) . For every \( s \in C, x \in X \), we have\n\n\[ s\left( x\right) \geq \inf \left( {{s}_{0}\left( x\right) , - {s}_{0}\left( {-x}\right) }\right) \]\n\nsince we ... | Yes |
Lemma 1.67. Let \( s \in S\left( X\right) \) and let \( u \in X \) . Then the function \( {s}_{u} \) given by\n\n\[ \n{s}_{u}\left( x\right) \mathrel{\text{:=}} \inf \left\{ {s\left( {x - {tu}}\right) - s\left( {-{tu}}\right) : t \in {\mathbb{R}}_{ + }}\right\} \n\]\n\nis sublinear and such that \( {s}_{u} \leq s,{s}_{... | Proof. For all \( x \in X \), the inequality \( {s}_{u}\left( x\right) \leq s\left( x\right) \) stems from the choice \( t = 0 \) in relation (1.16). Now, since for \( t \in {\mathbb{R}}_{ + } \) we have \( s\left( {-{tu}}\right) \leq s\left( {x - {tu}}\right) + s\left( {-x}\right) \), we get\n\n\[ \n\forall t \in {\ma... | Yes |
Corollary 1.68. For every \( s \in S\left( X\right) \) there exists some linear form \( \ell \) on \( X \) such that \( \ell \leq s \) . | Proof. Let \( {S}_{s}\left( X\right) \mathrel{\text{:=}} \left\{ {{s}^{\prime } \in S\left( X\right) : {s}^{\prime } \leq s}\right\} \) . The induced order on \( {S}_{s}\left( X\right) \) by \( S\left( X\right) \) is inductive, any chain \( C \) in \( {S}_{s}\left( X\right) \) being a chain in \( S\left( X\right) \) an... | Yes |
Corollary 1.69. Let \( X \) be a topological linear space and let \( h \) be a continuous sublinear function. Then there exists a continuous linear form \( \ell \) on \( X \) such that \( \ell \leq h \) . | Proof. It suffices to prove that if \( \ell \) is a linear form bounded above by \( h \), then \( \ell \) is continuous. Given \( \varepsilon > 0 \) we can find a symmetric neighborhood \( V \) of 0 such that \( h\left( v\right) \leq \varepsilon \) for every \( v \in V \) . Then for \( v \in V \), we have \( \ell \left... | Yes |
Lemma 1.70. Given a linear space \( X, h : X \rightarrow \mathbb{R}, k : X \rightarrow {\mathbb{R}}_{\infty } \) both sublinear and such that \( - k \leq h \), there exists some linear form \( \ell \) on \( X \) such that \( - k \leq \ell \leq h \) . | Proof. Let us introduce \( s : X \rightarrow \overline{\mathbb{R}} \) by\n\n\[ s\left( x\right) \mathrel{\text{:=}} \inf \{ h\left( {x + y}\right) + k\left( y\right) : y \in X\} . \]\n\nSince \( h\left( y\right) \leq h\left( {x + y}\right) + h\left( {-x}\right) \) and since \( h\left( y\right) \geq - k\left( y\right) \... | Yes |
Corollary 1.72. (a) Let \( X \) be a normed space and let \( \bar{x} \in X \) . Then there exists a continuous linear form \( \ell \) on \( X \) such that \( \parallel \ell \parallel \mathrel{\text{:=}} \sup \ell \left( {B}_{X}\right) \leq 1 \) and \( \ell \left( \bar{x}\right) = \parallel \bar{x}\parallel \) . | Proof. (a) Let \( {X}_{0} \mathrel{\text{:=}} \mathbb{R}\bar{x} \) and let \( {\ell }_{0} \) be the linear form on \( {X}_{0} \) given by \( {\ell }_{0}\left( {r\bar{x}}\right) = r\parallel \bar{x}\parallel \) for \( r \in \mathbb{R} \) . Thus, for every \( x \in {X}_{0} \), one has \( {\ell }_{0}\left( x\right) \leq h... | Yes |
Corollary 1.73. Let \( X \) be a normed space and let \( Y \) be a linear subspace of \( X \) . Then every continuous linear form \( {y}^{ * } \) on \( Y \) has a linear continuous extension \( {x}^{ * } \) to \( X \) such that \( \begin{Vmatrix}{x}^{ * }\end{Vmatrix} = \begin{Vmatrix}{y}^{ * }\end{Vmatrix} \) . | Proof. Let \( c \mathrel{\text{:=}} \begin{Vmatrix}{y}^{ * }\end{Vmatrix} \) . Theorem 1.71 provides some linear form \( \ell \) on \( X \) that extends \( {y}^{ * } \) and is bounded above by \( c\parallel \cdot \parallel \), hence is continuous. Clearly, \( \parallel \ell \parallel = \begin{Vmatrix}{y}^{ * }\end{Vmat... | Yes |
Corollary 1.75. Let \( Y \) be a closed linear subspace of a Banach space \( X \) . Then \( {Y}^{ * } \) is isometric to \( {X}^{ * }/{Y}^{ \bot } \), where \( {Y}^{ \bot } \mathrel{\text{:=}} \left\{ {{x}^{ * } \in {X}^{ * } : {x}^{ * }\left( y\right) = 0\forall y \in Y}\right\} \) . | Proof. Let \( r : {X}^{ * } \rightarrow {Y}^{ * } \) be the restriction map given by \( r\left( {x}^{ * }\right) \mathrel{\text{:=}} {\left. {x}^{ * }\right| }_{Y} \) . Corollary 1.73 ensures that \( r \) is onto. The kernel of \( r \) being precisely \( {Y}^{ \bot } \), one can factorize \( r \) as \( r = q \circ p \)... | Yes |
Corollary 1.76 (Sandwich theorem). Let \( X \) be a normed space, let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be a convex function, and let \( g \) be a concave function such that \( f \geq g \) . If \( g \) is continuous and finite at some point of the domain of \( f \), then there exists a continuous affine fu... | \[ f \geq h \geq g\text{.} \] | No |
Proposition 1.77. Let \( C \) be an absorbing convex subset of a linear space \( X \) and let \( e \in X \smallsetminus \operatorname{core}C \) . Then there exists a hyperplane \( H \) of \( X \) such that \( e \in H, H \cap \operatorname{core}C = \) \( \varnothing \) . Moreover, \( C \) is contained in one of the open... | Proof. Let \( j \mathrel{\text{:=}} {\mu }_{C} \) be the Minkowski gauge of \( C \) :\n\n\[ j\left( x\right) \mathrel{\text{:=}} \inf \{ t > 0 : x \in {tC}\} . \]\n\nSince \( C \) is absorbing and convex, \( j \) is finite on \( X \) and sublinear. For all \( x \in \operatorname{core}C \) one has \( j\left( x\right) < ... | Yes |
Theorem 1.78 (Eidelheit). Let \( A \) and \( B \) be two disjoint nonempty convex subsets of a topological linear space \( X \) . If \( A \) is open, then there exist some \( f \in {X}^{ * } \smallsetminus \{ 0\} \) , \( r \in \mathbb{R} \) such that\n\n\[ \forall a \in A,\forall b \in B,\;f\left( a\right) > r \geq f\l... | Proof. Let \( D \mathrel{\text{:=}} A - B \mathrel{\text{:=}} \{ a - b : a \in A, b \in B\} \) . It is a convex subset of \( X \) that is open as the union over \( b \in B \) of the translated sets \( A - b \), and \( 0 \notin D \) . Taking \( e \in D \) and setting \( C \mathrel{\text{:=}} e - D \), we see that \( e \... | Yes |
Theorem 1.79 (Hahn-Banach strong separation theorem). Let \( A \) and \( B \) be two disjoint nonempty convex subsets of a normed space (or a locally convex topological linear space) \( X \) . If \( A \) is compact and \( B \) is closed, then there exist some \( f \in {X}^{ * } \smallsetminus \{ 0\} \) and some \( r \i... | Proof. For every \( a \in A \) there exists a symmetric open convex neighborhood \( {V}_{a} \) of 0 in \( X \) such that \( \left( {a + 2{V}_{a}}\right) \cap B = \varnothing \) . Let \( F \) be a finite subset of \( A \) such that the family \( {\left( a + {V}_{a}\right) }_{a \in F} \) forms a finite covering of \( A \... | Yes |
Corollary 1.80 (Mazur). Every closed convex subset \( C \) of a normed space \( X \) is weakly closed. | Proof. It suffices to show that if \( C \) is a nonempty closed convex subset of \( X \) and if \( C \neq X \), then \( C \) coincides with the intersection of the family \( \mathcal{D} \) of closed half-spaces containing \( C \) . That amounts to showing that for \( a \in X \smallsetminus C \) there exists some \( D \... | Yes |
Corollary 1.81 (Hörmander). The map \( h : C \mapsto {h}_{C} \) is an injective map from the set \( \mathcal{C}\left( X\right) \) of nonempty closed convex subsets of the normed space \( X \) into the space \( \mathcal{H}\left( X\right) \) of positively homogeneous functions on \( X \) null at 0 . Moreover, \( {h}_{\la... | Proof. We just prove the injectivity of \( h \), leaving the other assertions as exercises (see \( \left\lbrack {{198},{591}}\right\rbrack \) ). It suffices to prove that for \( C, D \in \mathcal{C}\left( X\right) \) satisfying \( C \smallsetminus D \neq \varnothing \) one has \( {h}_{C} \neq {h}_{D} \) . Given \( b \i... | No |
Theorem 1.82 (Alaoglu-Bourbaki). Let \( X \) be a normed space and let \( S \) be a neighborhood of 0 in \( X \) . Then \( {S}^{0} \) is weak* compact. | Proof. Since \( {S}^{0} \subset {T}^{0} \) when \( T \subset S \) and since \( {S}^{0} \) is weak* closed, it suffices to prove the result when \( S \) is a ball centered at 0 . Since \( {\left( rS\right) }^{0} = {r}^{-1}{S}^{0} \) for \( r > 0 \), we may suppose \( S = {B}_{X} \) . Then \( {S}^{0} = {B}_{{X}^{ * }} \)... | Yes |
Corollary 1.83 (Bipolar theorem). For every nonempty subset \( S \) of a normed space \( X \), its bipolar is the closed convex hull of \( S \cup \{ 0\} : {S}^{00} \mathrel{\text{:=}} \overline{\mathrm{{co}}}\left( {S\cup \{ 0\} }\right) \) . | Proof. Let \( C \mathrel{\text{:=}} \overline{\operatorname{co}}\left( {S\cup \{ 0\} }\right) \) . Since one has \( S \subset {S}^{00} \), and since \( {S}^{00} \) is closed convex and contains 0, one has \( C \subset {S}^{00} \) . Given \( a \in X \smallsetminus C \), Theorem 1.79 yields \( {x}^{ * } \in {X}^{ * } \) ... | Yes |
Lemma 1.85. Let \( {f}_{1},\ldots ,{f}_{k} \) be convex functions on a convex subset \( C \) of a linear space \( X \) . Then\n\n\[ \mathop{\inf }\limits_{C}\left( {{f}_{1} \vee \cdots \vee {f}_{k}}\right) = \max \left\{ {\mathop{\inf }\limits_{C}\left( {{s}_{1}{f}_{1} + \cdots + {s}_{k}{f}_{k}}\right) : s \mathrel{\te... | Proof. For each \( s \mathrel{\text{:=}} \left( {{s}_{1},\ldots ,{s}_{k}}\right) \in {\Delta }_{k} \), we obviously have \( h \mathrel{\text{:=}} {f}_{1} \vee \cdots \vee {f}_{k} \geq {h}_{s} \mathrel{\text{:=}} \) \( {s}_{1}{f}_{1} + \cdots + {s}_{k}{f}_{k} \), hence \( \mathop{\inf }\limits_{C}h \geq {m}_{C}\left( s\... | Yes |
Theorem 1.86 (Infmax theorem). Let \( A \) and \( B \) be nonempty convex subsets of linear spaces \( X \) and \( Y \) respectively, and let \( \ell : A \times B \rightarrow \mathbb{R} \) be a function that is convex in its first variable and concave in its second variable. Then if \( B \) is compact for some topology ... | Proof. The inequality \( \alpha \mathrel{\text{:=}} \mathop{\inf }\limits_{{x \in A}}\mathop{\sup }\limits_{{y \in B}}\ell \left( {x, y}\right) \geq \beta \mathrel{\text{:=}} \mathop{\sup }\limits_{{y \in B}}\mathop{\inf }\limits_{{x \in A}}\ell \left( {x, y}\right) \) is valid without any assumption. Here we can write... | Yes |
Theorem 1.87 (Ekeland variational principle). Let \( \left( {X, d}\right) \) be a complete metric space and let \( f : X \rightarrow {\mathbb{R}}_{\infty } \mathrel{\text{:=}} \mathbb{R} \cup \{ + \infty \} \) be a bounded-below lower semicontinuous function with nonempty domain. Then for every \( \gamma > 0 \) one can... | \[ f\left( u\right) < f\left( x\right) + {\gamma d}\left( {u, x}\right) \;\text{ for all }x \in X \smallsetminus \{ u\} . \] | Yes |
Corollary 1.90 (Partial Ekeland theorem). Let \( \left( {W, d}\right) \) be a complete metric space, let \( X \) be a topological space, and let \( f : W \times X \rightarrow {\mathbb{R}}_{\infty } \) be a lower semicontinuous function with nonempty domain that is bounded below. Suppose that for every \( w \in W \) the... | Proof. Let \( p : W \rightarrow {\mathbb{R}}_{\infty } \) be the function given by \[ p\left( w\right) \mathrel{\text{:=}} \mathop{\inf }\limits_{{x \in X}}f\left( {w, x}\right) . \] It is bounded below, proper, and lower semicontinuous by Corollary 1.23. Let \( \left( {v}_{n}\right) \) be a sequence of \( W \) such th... | Yes |
Corollary 1.91. [906] Let \( \\left( {X, d}\\right) \) be a complete metric space and let \( f : X \\rightarrow {\\mathbb{R}}_{\\infty } \) be a bounded-below lower semicontinuous function. Suppose that for some \( \\gamma > 0 \) and for all \( w \\in X \) such that \( f\\left( w\\right) > \\inf f\\left( X\\right) \) t... | Proof. We may assume \( \\inf f\\left( X\\right) < + \\infty \) . Let \( u \\in X \) be given by Theorem 1.87. Assuming \( f\\left( u\\right) > \\inf f\\left( X\\right) \) and taking \( w = u \) in our assumption, we get a contradiction to relation (1.21). Thus \( f\\left( u\\right) = \\inf f\\left( X\\right) \) . | Yes |
Corollary 1.92. Let \( f : B \rightarrow \mathbb{R} \) be a lower semicontinuous function on an open ball \( B \mathrel{\text{:=}} B\left( {\bar{x}, r}\right) \) of a Banach space \( E \) . Suppose \( f \) is bounded below and Gâteaux differentiable on \( B \) . Then given \( \alpha > f\left( \bar{x}\right) - \inf f\le... | Proof. Let \( \varepsilon \in \left( {0,\alpha }\right) \) be such that \( \varepsilon > f\left( \bar{x}\right) - \inf f\left( B\right) \) and let \( \rho < \sigma < r \) be such that \( \rho > \varepsilon {\alpha }^{-1}r \) . Let us set \( X \mathrel{\text{:=}} B\left\lbrack {\bar{x},\sigma }\right\rbrack \) . Theorem... | Yes |
Corollary 1.93. Let \( E \) be a Banach space and let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be a bounded-below lower semicontinuous convex function on a ball \( B \mathrel{\text{:=}} B\left( {\bar{x}, r}\right) \) and finite at \( \bar{x} \) . Let \( \alpha > f\left( \bar{x}\right) - \inf f\left( B\right) \) .... | Proof. Let \( \varepsilon ,\rho ,\sigma, g \) be as in the preceding proof. Again we get a minimizer \( u \) of the function \( g \) belonging to the interior of \( B\left\lbrack {\bar{x},\sigma }\right\rbrack \), so that we have\n\n\[ \n{f}^{\prime }\left( {u, v}\right) + \varepsilon {\rho }^{-1}\parallel v\parallel \... | Yes |
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