Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Example 4.3.4 Let \( K = \mathbb{Q}\left( \sqrt{D}\right) \) with \( D \) a squarefree integer. Find an integral basis for \( {\mathcal{O}}_{K} \) . | Solution. An arbitrary element \( \alpha \) of \( K \) is of the form \( \alpha = {r}_{1} + {r}_{2}\sqrt{D} \) with \( {r}_{1},{r}_{2} \in \mathbb{Q} \) . Since \( \left\lbrack {K : \widehat{\mathbb{Q}}}\right\rbrack = 2,\alpha \) has only one conjugate: \( {r}_{1} - {r}_{2}\sqrt{D} \) . From Lemma 4.1.1 we know that i... | Yes |
Let \( K = \mathbb{Q}\left( \alpha \right) \) where \( \alpha = {r}^{1/3}, r = a{b}^{2} \in \mathbb{Z} \) where \( {ab} \) is squarefree. If \( 3 \mid r \), assume that \( 3 \mid a,3 \nmid b \) . Find an integral basis for \( K \) . | The minimal polynomial of \( \alpha \) is \( f\left( x\right) = {x}^{3} - r \), and \( \alpha \) ’s conjugates are \( \alpha ,{\omega \alpha } \), and \( {\omega }^{2}\alpha \) where \( \omega \) is a primitive cube root of unity. By Exercise 4.3.3,\n\n\[ \n{d}_{K/\mathbb{Q}}\left( \alpha \right) = - \mathop{\prod }\li... | No |
Theorem 5.1.2 Let \( R \) be a commutative ring with identity. Then:\n\n(a) \( \mathfrak{m} \) is a maximal ideal if and only if \( R/\mathfrak{m} \) is a field. | Proof. (a) By the correspondence between ideals of \( R \) containing \( \mathfrak{m} \) and ideals of \( R/\mathfrak{m}, R/\mathfrak{m} \) has a nontrivial ideal if and only if there is an ideal \( \mathfrak{a} \) of \( R \) strictly between \( \mathfrak{m} \) and \( R \) . Thus,\n\n\( \mathfrak{m} \) is maximal,\n\n\... | Yes |
Theorem 5.1.6 For \( \alpha \in \mathbb{C} \), the following are equivalent:\n\n(1) \( \alpha \) is integral over \( {\mathcal{O}}_{K} \) ;\n\n(2) \( {\mathcal{O}}_{K}\left\lbrack \alpha \right\rbrack \) is a finitely generated \( {\mathcal{O}}_{K} \) -module;\n\n(3) There is a finitely generated \( {\mathcal{O}}_{K} \... | Proof. \( \left( 1\right) \Rightarrow \left( 2\right) \) Let \( \alpha \in \mathbb{C} \) be integral over \( {\mathcal{O}}_{K} \) . Say \( \alpha \) satisfies a monic polynomial of degree \( n \) over \( {\mathcal{O}}_{K} \) . Then\n\n\[{\mathcal{O}}_{K}\left\lbrack \alpha \right\rbrack = {\mathcal{O}}_{K} + {\mathcal{... | Yes |
Theorem 5.1.7 \( {\mathcal{O}}_{K} \) is integrally closed. | Proof. If \( \alpha \in K \) is integral over \( {\mathcal{O}}_{K} \), then let\n\n\[ M = {\mathcal{O}}_{K}{u}_{1} + \cdots + {\mathcal{O}}_{K}{u}_{n},\;{\alpha M} \subseteq M.\]\n\nLet \( {\mathcal{O}}_{K} = \mathbb{Z}{v}_{1} + \cdots + \mathbb{Z}{v}_{m} \), where \( \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \) is a b... | Yes |
Theorem 5.2.3 For any commutative ring \( R \) with identity, the following are equivalent:\n\n(1) \( R \) is Noetherian;\n\n(2) every nonempty set of ideals contains a maximal element; and\n\n(3) every ideal of \( R \) is finitely generated. | Proof. (1) \( \Rightarrow \) (2) Suppose that \( S \) is a nonempty set of ideals of \( R \) that does not contain a maximal element. Let \( {\mathfrak{a}}_{1} \in S.{\mathfrak{a}}_{1} \) is not maximal in \( S \), so there is an \( {\mathfrak{a}}_{2} \in S \) with \( {\mathfrak{a}}_{1} \varsubsetneq {\mathfrak{a}}_{2}... | Yes |
Lemma 5.3.3 Any proper ideal of \( {\mathcal{O}}_{K} \) contains a product of nonzero prime ideals. | Proof. Let \( S \) be the set of all proper ideals of \( {\mathcal{O}}_{K} \) that do not contain a product of prime ideals. We need to show that \( S \) is empty. If not, then since \( {\mathcal{O}}_{K} \) is Noetherian, \( S \) has a maximal element, say \( \mathfrak{a} \) . Then, \( \mathfrak{a} \) is not prime sinc... | Yes |
Lemma 5.3.4 Let \( \wp \) be a prime ideal of \( {\mathcal{O}}_{K} \). There exists \( z \in K, z \notin {\mathcal{O}}_{K} \), such that \( {z}_{\wp } \subseteq {\mathcal{O}}_{K} \). | Proof. Take \( x \in \wp \). From the previous lemma, \( \left( x\right) = x{\mathcal{O}}_{K} \) contains a product of prime ideals. Let \( r \) be the least integer such that \( \left( x\right) \) contains a product of \( r \) prime ideals, and say \( \left( x\right) \supseteq {\wp }_{1}\cdots {\wp }_{r} \), with the ... | Yes |
Theorem 5.3.5 Let \( \wp \) be a prime ideal of \( {\mathcal{O}}_{K} \) . Then \( {\wp }^{-1} \) is a fractional ideal and \( \wp {\wp }^{-1} = {\mathcal{O}}_{K} \) . | Proof. It is easily seen that \( {\wp }^{-1} \) is an \( {\mathcal{O}}_{K} \) -module. Now, \( \wp \cap \mathbb{Z} \neq \left( 0\right) \), from Exercise 4.4.1, so let \( n \in \wp \cap \mathbb{Z}, n \neq 0 \) . Then, \( n{\wp }^{-1} \subseteq \wp {\wp }^{-1} \subseteq {\mathcal{O}}_{K} \), by definition. Thus, \( {\wp... | Yes |
Theorem 5.3.6 Any ideal of \( {\mathcal{O}}_{K} \) can be written as a product of prime ideals uniquely. | ## Proof.\n\nExistence. Let \( S \) be the set of ideals of \( {\mathcal{O}}_{K} \) that cannot be written as a product of prime ideals. If \( S \) is nonempty, then \( S \) has a maximal element, since \( {\mathcal{O}}_{K} \) is Noetherian. Let \( \mathfrak{a} \) be a maximal element of \( S \) . Then \( \mathfrak{a} ... | Yes |
Theorem 5.3.13 (Chinese Remainder Theorem) (a) Let \( \mathfrak{a},\mathfrak{b} \) be ideals so that \( \gcd \left( {\mathfrak{a},\mathfrak{b}}\right) = 1 \), i.e., \( \mathfrak{a} + \mathfrak{b} = {\mathcal{O}}_{K} \) . Given \( a, b \in {\mathcal{O}}_{K} \), we can solve\n\n\[ x \equiv a\;\left( {\;\operatorname{mod}... | Proof. (a) Since \( \mathfrak{a} + \mathfrak{b} = {\mathcal{O}}_{K},\exists {x}_{1} \in \mathfrak{a},{x}_{2} \in \mathfrak{b} \) with \( {x}_{1} + {x}_{2} = 1 \) . Let \( x = b{x}_{1} + a{x}_{2} \equiv a{x}_{2}\left( {\;\operatorname{mod}\;\mathfrak{a}}\right) \) . But, \( {x}_{2} = 1 - {x}_{1} \equiv 1\left( {\;\opera... | Yes |
Theorem 5.3.16 (a) If \( \mathfrak{a} = \mathop{\prod }\limits_{{i = 1}}^{r}{\wp }_{i}^{{e}_{i}} \), then\n\n\[ N\left( \mathfrak{a}\right) = \mathop{\prod }\limits_{{i = 1}}^{r}N\left( {\wp }_{i}^{{e}_{i}}\right) . \] | Proof. (a) Consider the map\n\n\[ \phi : {\mathcal{O}}_{K} \rightarrow {\bigoplus }_{i = 1}^{r}\left( {{\mathcal{O}}_{K}/{\wp }_{i}^{{e}_{i}}}\right) \]\n\n\[ x \rightarrow \left( {{x}_{1},\ldots ,{x}_{r}}\right) \]\n\nwhere \( {x}_{i} \equiv x\left( {\;\operatorname{mod}\;{\wp }_{i}^{{e}_{i}}}\right) \). \n\nThe funct... | Yes |
Theorem 5.4.3 Let \( \mathcal{D} \) be the different of an algebraic number field \( K \) . Then \( N\left( \mathcal{D}\right) = \left| {d}_{K}\right| \) . | Proof. For some \( m > 0, m{\mathcal{D}}^{-1} \) is an ideal of \( {\mathcal{O}}_{K} \) . Now,\n\n\[ m{\mathcal{D}}^{-1} = \mathbb{Z}m{\omega }_{1}^{ * } + \cdots + \mathbb{Z}m{\omega }_{n}^{ * } \]\n\nLet\n\n\[ m{\omega }_{i}^{ * } = \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{ij}{\omega }_{j} \]\n\nso\n\n\[ {\omega }_{i}... | Yes |
Theorem 5.4.4 Let \( p \in \mathbb{Z} \) be prime, \( \wp \subseteq {\mathcal{O}}_{K} \), a prime ideal and \( \mathcal{D} \) the different of \( K \) . If \( {\wp }^{e} \mid \left( p\right) \), then \( {\wp }^{e - 1} \mid \mathcal{D} \). | Proof. We may assume that \( e \) is the highest power of \( \wp \) dividing \( \left( p\right) \). So let \( \left( p\right) = {\wp }^{e}\mathfrak{a},\gcd \left( {\mathfrak{a},\wp }\right) = 1 \). Let \( x \in \wp \mathfrak{a} \). Then \( x = \mathop{\sum }\limits_{{i = 1}}^{n}{p}_{i}{a}_{i},{p}_{i} \in \wp ,{a}_{i} \... | Yes |
Theorem 6.2.2 There exists a constant \( {C}_{K} \) such that every ideal \( \mathfrak{a} \subseteq {\mathcal{O}}_{K} \) is equivalent to an ideal \( \mathfrak{b} \subseteq {\mathcal{O}}_{K} \) with \( N\left( \mathfrak{b}\right) \leq {C}_{K} \) . | Proof. Suppose \( \mathfrak{a} \) is an ideal of \( {\mathcal{O}}_{K} \) . Let \( \beta \in \mathfrak{a} \) be a non-zero element such that \( \left| {N\left( \beta \right) }\right| \) is minimal.\n\nFor each \( \alpha \in \mathfrak{a} \), by Exercise 6.1.2, we can find \( t \in \mathbb{Z},\left| t\right| \leq {H}_{K} ... | No |
Theorem 6.2.5 The number of equivalence classes of ideals is finite. | Proof. By Exercise 6.2.3, each equivalence class of ideals can be represented by an integral ideal. This integral ideal, by Theorem 6.2.2, is equivalent to another integral ideal with norm less than or equal to a given constant \( {C}_{K} \) . Apply Exercise 6.2.4, and we are done. | No |
Show that the equation \( {x}^{2} + 5 = {y}^{3} \) has no integral solution. | Observe that if \( y \) is even, then \( x \) is odd, and \( {x}^{2} + 5 \equiv 0\left( {\;\operatorname{mod}\;4}\right) \), and hence \( {x}^{2} \equiv 3\left( {\;\operatorname{mod}\;4}\right) \), which is a contradiction. Therefore, \( y \) is odd. Also, if a prime \( p \mid \left( {x, y}\right) \), then \( p \mid 5 ... | Yes |
Theorem 7.3.1 (Law of Quadratic Reciprocity) Let \( p \) and \( q \) be odd primes. Then\n\n\[ \left( \frac{p}{q}\right) = \left( \frac{q}{p}\right) {\left( -1\right) }^{\frac{p - 1}{2} \cdot \frac{q - 1}{2}} \] | Proof. From Exercise 7.2.2, we have\n\n\[ {S}^{q} \equiv \left( \frac{q}{p}\right) S\;\left( {\;\operatorname{mod}\;q}\right) \]\n\nThus, cancelling out an \( S \) from both sides will give us\n\n\[ {S}^{q - 1} \equiv \left( \frac{q}{p}\right) \;\left( {\;\operatorname{mod}\;q}\right) \]\n\nSince \( q \) is odd, \( q -... | No |
Theorem 7.4.5 Suppose \( p = 2 \) . Then:\n\n(a) \( 2{\mathcal{O}}_{K} = {\wp }^{2} \) , \( \wp \) prime if and only if \( 2 \mid {d}_{K} \) ; | Proof. (a) \( \Leftarrow \) If \( 2 \mid {d}_{K} \), then \( d \equiv 2,3\left( {\;\operatorname{mod}\;4}\right) \) . If \( d \equiv 2\left( {\;\operatorname{mod}\;4}\right) \), then we claim that \( \left( 2\right) = {\left( 2,\sqrt{d}\right) }^{2} \) . Note that\n\n\[{\left( 2,\sqrt{d}\right) }^{2} = \left( {4,2\sqrt... | Yes |
Lemma 8.1.4 (a) Let \( m, n \in \mathbb{Z} \) with \( 0 < m \leq n \) and let \( \Delta = \left( {d}_{ij}\right) \in \) \( {M}_{n \times m}\left( \mathbb{R}\right) \) . For any integer \( t > 1 \), there is a nonzero \( \mathbf{x} = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in \) \( {\mathbb{Z}}^{n} \) with each \( \lef... | Proof. (a) Let \( \delta = \mathop{\max }\limits_{{1 \leq j \leq m}}\mathop{\sum }\limits_{{i = 1}}^{n}\left| {d}_{ij}\right| \) . Then, for\n\n\[ \mathbf{0} \neq \mathbf{x} = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in {\mathbb{Z}}_{ \geq 0}^{n} \]\n\nwith each \( \left| {x}_{i}\right| \leq t \) ,\n\n\[ \left| {y}_{j}... | Yes |
Lemma 8.1.5 Let \( E = A \cup B \) be a proper partition of \( E \). (a) There exists a sequence of nonzero integers \( \\left\\{ {\\alpha }_{v}\\right\\} \\subseteq {\\mathcal{O}}_{K} \) such that \[ \\left| {\\alpha }_{v}^{\\left( k\\right) }\\right| > \\left| {\\alpha }_{v + 1}^{\\left( k\\right) }\\right| \\;\\text... | Proof. (a) Let \( {t}_{1} \) be an integer greater than 1 and let \( \\left\\{ {t}_{v}\\right\\} \) be the sequence defined recursively by the relation \( {t}_{v + 1} = M{t}_{v} \) for all \( v \\geq 1 \), where \( M \) is a positive constant that will be suitably chosen. By Lemma 8.1.4, for each \( v,\\exists {\\alpha... | Yes |
Theorem 8.1.10 (a) Let \( {\zeta }_{m} = {e}^{{2\pi i}/m}, K = \mathbb{Q}\left( {\zeta }_{m}\right) \) . If \( m \) is even, the only roots of unity in \( K \) are the mth roots of unity, so that \( {W}_{K} \simeq \mathbb{Z}/m\mathbb{Z} \) . If \( m \) is odd, the only ones are the \( {2m} \) th roots of unity, so that... | Proof. (a) If \( m \) is odd, then \( {\zeta }_{2m} = - {\zeta }_{2m}^{m + 1} = - {\zeta }_{m}^{\left( {m + 1}\right) /2} \) which implies that \( \mathbb{Q}\left( {\zeta }_{m}\right) = \mathbb{Q}\left( {\zeta }_{2m}\right) \) . It will, therefore, suffice to establish the statement for \( m \) even. Suppose that \( \t... | Yes |
Theorem 8.2.4 (a) Let \( \alpha \) be an irrational number and let \( {C}_{j} = {p}_{j}/{q}_{j} \) , for \( j \in \mathbb{N} \), be the convergents of the simple continued fraction of \( \alpha \) . If \( r, s \in \mathbb{Z} \) with \( s > 0 \) and \( k \) is a positive integer such that\n\n\[ \left| {{s\alpha } - r}\r... | Proof. (a) Suppose, on the contrary, that \( 1 \leq s < {q}_{k + 1} \) . For each \( k \geq 0 \) , consider the system of linear equations\n\n\[ {p}_{k}x + {p}_{k + 1}y = r \]\n\n\[ {q}_{k}x + {q}_{k + 1}y = s. \]\n\nUsing Gaussian elimination, we easily find that\n\n\[ \left( {{p}_{k + 1}{q}_{k} - {p}_{k}{q}_{k + 1}}\... | Yes |
Theorem 9.1.10 If \( \chi \neq {\chi }_{0} \), then \( \left| {g\left( \chi \right) }\right| = \sqrt{p} \). | Proof. We first observe that \( a \neq 0 \) and \( \chi \neq {\chi }_{0} \) together imply that \( {g}_{a}\left( \chi \right) = \chi \left( {a}^{-1}\right) g\left( \chi \right) \) because\n\n\[ \chi \left( a\right) {g}_{a}\left( \chi \right) = \chi \left( a\right) \mathop{\sum }\limits_{{t \in {\mathbb{F}}_{p}}}\chi \l... | Yes |
Lemma 9.1.13 Let \( \pi \) be a prime of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) such that \( N\left( \pi \right) = p \equiv 1\left( {\;\operatorname{mod}\;3}\right) \) . The character \( {\chi }_{\pi } \) introduced above can be viewed as a character of the finite field \( \mathbb{Z}\left\lbrack \rho \right\rb... | Proof. If \( \chi \) is any cubic character, Exercise 9.1.12 shows that \( g{\left( \chi \right) }^{3} = \) \( {pJ}\left( {\chi ,\chi }\right) \) since \( \chi \left( {-1}\right) = 1 \) . We can write \( J\left( {\chi ,\chi }\right) = a + {b\rho } \) for some \( a, b \in \mathbb{Z} \) . But\n\n\[ g{\left( \chi \right) ... | Yes |
Lemma 9.1.15 Let \( {\pi }_{1} = q \equiv 2\\left( {\\;\\operatorname{mod}\\;3}\\right) \) and \( {\pi }_{2} = \\pi \) be a prime of \( \\mathbb{Z}\\left\\lbrack \\rho \\right\\rbrack \) of norm \( p \) . Then \( {\\chi }_{\\pi }\\left( q\\right) = {\\chi }_{q}\\left( \\pi \\right) \) . In other words,\n\n\\[ \n{\\left... | Proof. Let \( {\\chi }_{\\pi } = \\chi \), and consider the Jacobi sum \( J\\left( {\\chi ,\\ldots ,\\chi }\\right) \) with \( q \) terms. Since \( 3 \\mid q + 1 \), we have, by Exercise 9.1.12, \( g{\\left( \\chi \\right) }^{q + 1} = {pJ}\\left( {\\chi ,\\ldots ,\\chi }\\right) \) . By Exercise 9.1.14, \( g{\\left( \\... | No |
Theorem 9.1.16 Let \( {\pi }_{1} \) and \( {\pi }_{2} \) be two primary primes of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \), of norms \( {p}_{1},{p}_{2} \), respectively. Then \( {\chi }_{{\pi }_{1}}\left( {\pi }_{2}\right) = {\chi }_{{\pi }_{2}}\left( {\pi }_{1}\right) \). In other words,\n\n\[ \n{\left( \frac{{... | Proof. To begin, let \( {\gamma }_{1} = \overline{{\pi }_{1}},{\gamma }_{2} = \overline{{\pi }_{2}} \). Then \( {p}_{1} = {\pi }_{1}{\gamma }_{1},{p}_{2} = {\pi }_{2}{\gamma }_{2} \), and \( {p}_{1},{p}_{2} \equiv 1\left( {\;\operatorname{mod}\;3}\right) \). Now,\n\n\[ \ng{\left( {\chi }_{{\gamma }_{1}}\right) }^{{p}_{... | No |
Theorem 10.1.4 Let \( {\left\{ {a}_{m}\right\} }_{m = 1}^{\infty } \) be a sequence of complex numbers, and let \( A\left( x\right) = \mathop{\sum }\limits_{{m \leq x}}{a}_{m} = O\left( {x}^{\delta }\right) \), for some \( \delta \geq 0 \) . Then\n\n\[ \mathop{\sum }\limits_{{m = 1}}^{\infty }\frac{{a}_{m}}{{m}^{s}} \]... | Proof. We write\n\n\[ \mathop{\sum }\limits_{{m = 1}}^{M}\frac{{a}_{m}}{{m}^{s}} = \mathop{\sum }\limits_{{m = 1}}^{M}\left( {A\left( m\right) - A\left( {m - 1}\right) }\right) {m}^{-s} \]\n\n\[ = A\left( M\right) {M}^{-s} + \mathop{\sum }\limits_{{m = 1}}^{{M - 1}}A\left( m\right) \left\{ {{m}^{-s} - {\left( m + 1\rig... | Yes |
Let \( K = \mathbb{Q}\left( i\right) \). Show that \( \left( {s - 1}\right) {\zeta }_{K}\left( s\right) \) extends to an analytic function for \( \operatorname{Re}\left( s\right) > \frac{1}{2} \). | Solution. Since every ideal \( \mathfrak{a} \) of \( {\mathcal{O}}_{K} \) is principal, we can write \( \mathfrak{a} = \left( {a + {ib}}\right) \) for some integers \( a, b \). Moreover, since\n\n\[ \mathfrak{a} = \left( {a + {ib}}\right) = \left( {-a - {ib}}\right) = \left( {-a + {ib}}\right) = \left( {a - {ib}}\right... | Yes |
Theorem 10.2.8 (Dirichlet's Hyperbola Method) Let\n\n\\[ \nf\left( n\right) = \mathop{\sum }\limits_{{\delta \mid n}}g\left( \delta \right) h\left( \frac{n}{\delta }\right) \n\\]\n\nand define\n\n\\[ \nG\left( x\right) = \mathop{\sum }\limits_{{n \leq x}}g\left( n\right) \n\\]\n\n\\[ \nH\left( x\right) = \mathop{\sum }... | Proof. We have\n\n\\[ \n\mathop{\sum }\limits_{{n \leq x}}f\left( n\right) = \mathop{\sum }\limits_{{{\delta e} \leq x}}g\left( \delta \right) h\left( e\right) \n\\]\n\n\\[ \n= \mathop{\sum }\limits_{\substack{{{\delta e} \leq x} \\ {\delta \leq y} }}g\left( \delta \right) h\left( e\right) + \mathop{\sum }\limits_{\sub... | Yes |
Let \( K \) be a quadratic field, and \( {a}_{n} \) the number of ideals of norm \( n \) in \( {\mathcal{O}}_{K} \) . Show that\n\n\[ \mathop{\sum }\limits_{{n \leq x}}{a}_{n} = {cx} + O\left( \sqrt{x}\right) \]\n\nwhere\n\n\[ c = \mathop{\sum }\limits_{{\delta = 1}}^{\infty }\left( \frac{{d}_{K}}{\delta }\right) \frac... | Solution. By Exercise 10.2.5,\n\n\[ {a}_{n} = \mathop{\sum }\limits_{{\delta \mid n}}\left( \frac{{d}_{K}}{\delta }\right) \]\n\nso that we can apply Theorem 10.2.8 with \( g\left( \delta \right) = \left( \frac{{d}_{K}}{\delta }\right) \) and \( h\left( \delta \right) = 1 \) , \( y = \sqrt{x} \) . We get\n\n\[ \mathop{... | No |
Lemma 10.4.1 Let \( \\left\\{ {a}_{n}\\right\\} \) be a sequence of nonnegative numbers. There exists a \( {\\sigma }_{0} \\in \\mathbb{R} \) (possibly infinite) such that\n\n\[ f\\left( s\\right) = \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }\\frac{{a}_{n}}{{n}^{s}} \]\n\nconverges for \( \\sigma > {\\sigma }_{0} \) ... | Proof. If there is no real value of \( s \) for which the series converges, we take \( {\\sigma }_{0} = \\infty \) . Therefore, suppose there is some real \( {s}_{0} \) for which the series converges. Clearly by the comparison test, the series converges for \( \\operatorname{Re}\\left( s\\right) > {s}_{0} \) since the ... | Yes |
Theorem 10.4.2 Let \( {a}_{n} \geq 0 \) be a sequence of nonnegative numbers. Then\n\n\[ f\left( s\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}_{n}}{{n}^{s}} \]\n\ndefines a holomorphic function in \( \operatorname{Re}\left( s\right) > {\sigma }_{0} \) and \( s = {\sigma }_{0} \) is a singular point of \... | Proof. By the previous lemma, it is clear that \( f\left( s\right) \) is holomorphic in \( \operatorname{Re}\left( s\right) > {\sigma }_{0} \) . If \( f \) is not singular at \( s = {\sigma }_{0} \), then there is a disk\n\n\[ D = \left\{ {s : \left| {s - {\sigma }_{1}}\right| < \delta }\right\} \]\n\nwhere \( {\sigma ... | Yes |
Theorem 10.4.5 Let \( L\\left( {s,\\chi }\\right) \) be defined as above. Then \( L\\left( {1,\\chi }\\right) \\neq 0 \) for \( \\chi \\neq {\\chi }_{0} \) . | Proof. By the previous exercise, the abscissa of convergence of\n\n\\[ \nf\\left( s\\right) = \\mathop{\\prod }\\limits_{\\chi }L\\left( {s,\\chi }\\right) = \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }\\frac{{c}_{n}}{{n}^{s}} \\]\n\nis greater than or equal to \( 1/\\varphi \\left( m\\right) \) . If for some \( \\chi... | Yes |
Theorem 11.2.4 Let \( n \) be the degree of \( K/\mathbb{Q} \) . If \( \chi \neq {\chi }_{0} \), then \( L\left( {s,\chi }\right) \) extends analytically to \( \Re \left( s\right) > 1 - \frac{1}{n} \) . | Proof. By Theorem 11.1.5, we have\n\n\[ \mathop{\sum }\limits_{C}\mathop{\sum }\limits_{{\mathfrak{a} \in C, N\left( \mathfrak{a}\right) \leq x}}\chi \left( \mathfrak{a}\right) = \mathop{\sum }\limits_{C}\chi \left( C\right) N\left( {x, C}\right) = O\left( {x}^{1 - \frac{1}{n}}\right) ,\]\n\nsince (by Exercise 11.2.2)\... | No |
Proposition 2.3. Let \( \mathbf{E} \) be a Banach space and \( x \neq 0 \) an element of \( \mathbf{E} \) . Then there exists a continuous linear map \( \lambda \) of \( \mathbf{E} \) into \( \mathbf{R} \) such that \( \lambda \left( x\right) \neq 0 \) . | One constructs \( \lambda \) by Zorn’s lemma, supposing that \( \lambda \) is defined on some subspace, and having a bounded norm. One then extends \( \lambda \) to the subspace generated by one additional element, without increasing the norm. | No |
Proposition 2.5. Let \( \\mathbf{E},\\mathbf{F} \) be two Banach spaces. Then the set of toplinear isomorphisms \( \\operatorname{Lis}\\left( {\\mathbf{E},\\mathbf{F}}\\right) \) is open in \( L\\left( {\\mathbf{E},\\mathbf{F}}\\right) \) . | The proof is in fact quite simple. If \( \\operatorname{Lis}\\left( {\\mathbf{E},\\mathbf{F}}\\right) \) is not empty, one is immediately reduced to proving that \( \\operatorname{Laut}\\left( \\mathbf{E}\\right) \) is open in \( L\\left( {\\mathbf{E},\\mathbf{E}}\\right) \). We then remark that if \( u \\in L\\left( {... | No |
Proposition 3.1. If \( f : U \rightarrow V \) is differentiable at \( {x}_{0} \), if \( g : V \rightarrow W \) is differentiable at \( f\left( {x}_{0}\right) \), then \( g \circ f \) is differentiable at \( {x}_{0} \), and\n\n\[{\left( g \circ f\right) }^{\prime }\left( {x}_{0}\right) = {g}^{\prime }\left( {f\left( {x}... | Proof. We leave it as a simple (and classical) exercise. | No |
Proposition 3.4. Let \( U \) be open in \( \mathbf{E} \), and let \( {f}_{i} : U \rightarrow {\mathbf{F}}_{i}\left( {i = 1,\ldots, n}\right) \) be continuous maps into spaces \( {\mathbf{F}}_{i} \) . Let \( f = \left( {{f}_{1},\ldots ,{f}_{n}}\right) \) be the map of \( U \) into the product of the \( {\mathbf{F}}_{i} ... | \[ {D}^{p}f = \left( {{D}^{p}{f}_{1},\ldots ,{D}^{p}{f}_{n}}\right) . \] | Yes |
Proposition 3.5. Let \( {U}_{1},\ldots ,{U}_{n} \) be open in the spaces \( {\mathbf{E}}_{1},\ldots ,{\mathbf{E}}_{n} \) and let \( f : {U}_{1} \times \cdots \times {U}_{n} \rightarrow \mathbf{F} \) be a continuous map. Then \( f \) is of class \( {C}^{p} \) if and only if each partial derivative \( {D}_{i}f : {U}_{1} ... | \[ v = \left( {{v}_{1},\ldots ,{v}_{n}}\right) \in {\mathbf{E}}_{1} \times \cdots \times {\mathbf{E}}_{n} \] we have \[ {Df}\left( x\right) \cdot \left( {{v}_{1},\ldots ,{v}_{n}}\right) = \sum {D}_{i}f\left( x\right) \cdot {v}_{i}. \] | Yes |
Proposition 3.7. Let \( \\mathbf{E},\\mathbf{F},\\mathbf{G} \) be Banach spaces, and \( U \) open in \( \\mathbf{E} \) . Let \( f : U \\rightarrow \\mathbf{F} \) be of class \( {C}^{p} \) and \( g : \\mathbf{F} \\rightarrow \\mathbf{G} \) continuous and linear. Then \( g \\circ f \) is of class \( {C}^{p} \) and | \[ {D}^{p}\\left( {g \\circ f}\\right) = g \\circ {D}^{p}f. \] | Yes |
Proposition 3.9. Let \( \\mathbf{E},\\mathbf{F} \) be Banach spaces which are toplinearly isomorphic. If \( u : \\mathbf{E} \\rightarrow \\mathbf{F} \) is a toplinear isomorphism, we denote its inverse by \( {u}^{-1} \). Then the map | \[ u \\mapsto {u}^{-1} \] from \( \\operatorname{Lis}\\left( {\\mathbf{E},\\mathbf{F}}\\right) \) to \( \\operatorname{Lis}\\left( {\\mathbf{F},\\mathbf{E}}\\right) \) is a \( {C}^{\\infty } \)-isomorphism. Its derivative at a point \( {u}_{0} \) is the linear map of \( L\\left( {\\mathbf{E},\\mathbf{F}}\\right) \) int... | Yes |
Proposition 3.10. Let \( U \) be open in the Banach space \( \mathbf{E} \) and let \( \mathbf{F},\mathbf{G} \) be Banach spaces.\n\n(i) If \( f : U \rightarrow L\left( {\mathbf{E},\mathbf{F}}\right) \) is a \( {C}^{p} \) -morphism, then the map of \( U \times \mathbf{E} \) into F given by\n\n\[ \left( {x, v}\right) \ma... | This proposition concludes our summary of results assumed without proof. | No |
Proposition 4.1. Let \( \lambda : \mathbf{E} \rightarrow \mathbf{R} \) be a continuous linear map and let \( f : I \rightarrow \mathbf{E} \) be ruled. Then \( {\lambda f} = \lambda \circ f \) is ruled, and\n\n\[ \lambda {\int }_{a}^{b}f\left( t\right) {dt} = {\int }_{a}^{b}{\lambda f}\left( t\right) {dt} \] | Proof. If \( {f}_{n} \) is a sequence of step functions converging uniformly to \( f \) , then \( \lambda {f}_{n} \) is ruled and converges uniformly to \( {\lambda f} \) . Our formula follows at once. | Yes |
Corollary 4.2. Let \( \\mathbf{E} \) , \( \\mathbf{F} \) be two Banach spaces, \( U \) open in \( \\mathbf{E} \), and \( x, z \) two distinct points of \( U \) such that the segment \( x + t\\left( {z - x}\\right) \\left( {0 \\leqq t \\leqq 1}\\right) \) lies in U. Let \( f : U \\rightarrow \\mathbf{F} \) be continuous... | Proof. This comes from the usual estimations of the integral. Indeed, for any continuous map \( g : I \\rightarrow \\mathbf{F} \) we have the estimate\n\n\[ \n\\left| {{\\int }_{a}^{b}g\\left( t\\right) {dt}}\\right| \\leqq K\\left( {b - a}\\right)\n\]\n\nif \( K \) is a bound for \( g \) on \( I \), and \( a \\leqq b ... | Yes |
Corollary 4.3. Let the hypotheses be as in Corollary 4.2. Let \( {x}_{0} \) be a point on the segment between \( x \) and \( z \) . Then\n\n\[ \left| {f\left( z\right) - f\left( x\right) - {f}^{\prime }\left( {x}_{0}\right) \left( {z - x}\right) }\right| \leqq \left| {z - x}\right| \sup \left| {{f}^{\prime }\left( \xi ... | Proof. We apply Corollary 4.2 to the map\n\n\[ g\left( x\right) = f\left( x\right) - {f}^{\prime }\left( {x}_{0}\right) x. \] | Yes |
Lemma 5.1 (Contraction Lemma or Shrinking Lemma). Let \( M \) be a complete metric space, with distance function \( d \), and let \( f : M \rightarrow M \) be a mapping of \( M \) into itself. Assume that there is a constant \( K,0 < K < 1 \) , such that, for any two points \( x, y \) in \( M \), we have\n\n\[ d\left( ... | Proof. This is a trivial exercise in the convergence of the geometric series, which we leave to the reader. | No |
Lemma 5.4. Let \( U \) be open in \( \mathbf{E} \), and let \( f : U \rightarrow \mathbf{E} \) be of class \( {C}^{1} \) . Assume that \( f\left( 0\right) = 0,{f}^{\prime }\left( 0\right) = I \) . Let \( r > 0 \) and assume that \( {\bar{B}}_{r}\left( 0\right) \subset U \) . Let \( 0 < s < 1 \), and assume that\n\n\[ \... | Proof. The map \( {g}_{y} \) given by \( {g}_{y}\left( x\right) = x - f\left( x\right) + y \) is defined for \( \left| x\right| \leqq r \) and \( \left| y\right| \leqq \left( {1 - s}\right) r \), and maps \( {\bar{B}}_{r}\left( 0\right) \) into itself because, from the estimate\n\n\[ \left| {f\left( x\right) - x}\right... | Yes |
Corollary 5.5. Let \( U \) be an open subset of \( \mathbf{E} \), and \( f : U \rightarrow {\mathbf{F}}_{1} \times {\mathbf{F}}_{2}a \) morphism of \( U \) into a product of Banach spaces. Let \( {x}_{0} \in U \), suppose that \( f\left( {x}_{0}\right) = \left( {0,0}\right) \) and that \( {f}^{\prime }\left( {x}_{0}\ri... | Proof. We may assume without loss of generality that \( {\mathbf{F}}_{1} = \mathbf{E} \) (identify by means of \( {f}^{\prime }\left( {x}_{0}\right) \) ) and \( {x}_{0} = 0 \) . We define\n\n\[ \varphi : U \times {\mathbf{F}}_{2} \rightarrow {\mathbf{F}}_{1} \times {\mathbf{F}}_{2} \]\n\nby the formula\n\n\[ \varphi \l... | Yes |
Corollary 5.7. Let \( U \) be an open subset of a product of Banach spaces \( {\mathbf{E}}_{1} \times {\mathbf{E}}_{2} \) and \( \left( {{a}_{1},{a}_{2}}\right) \) a point of \( U \) . Let \( f : U \rightarrow \mathbf{F} \) be a morphism into a Banach space, say \( f\left( {{a}_{1},{a}_{2}}\right) = 0 \), and assume th... | Proof. We may assume \( \left( {{a}_{1},{a}_{2}}\right) = \left( {0,0}\right) \) and \( {\mathbf{E}}_{2} = \mathbf{F} \) . We define\n\n\[ \n\varphi : {\mathbf{E}}_{1} \times {\mathbf{E}}_{2} \rightarrow {\mathbf{E}}_{1} \times {\mathbf{E}}_{2} \n\]\n\nby\n\n\[ \n\varphi \left( {{x}_{1},{x}_{2}}\right) = \left( {{x}_{1... | Yes |
Corollary 5.8. Let \( U \) be an open subset of a Banach space \( \mathbf{E} \) and \( f : U \rightarrow \mathbf{F} \) a morphism into a Banach space \( \mathbf{F} \). Let \( {x}_{0} \in U \) and assume that \( {f}^{\prime }\left( {x}_{0}\right) \) is surjective, and that its kernel splits. Then there exists an open su... | Proof. Again this is essentially a reformulation of the corollary, taking into account the splitting assumption. | No |
Lemma 2.1. Let \( {U}_{1},{U}_{2},{V}_{1},{V}_{2} \) be open subsets of Banach spaces, and \( g : {U}_{1} \times {U}_{2} \rightarrow {V}_{1} \times {V}_{2} \) a \( {C}^{p} \) -morphism. Let \( {a}_{2} \in {U}_{2} \) and \( {b}_{2} \in {V}_{2} \) and assume that \( g \) maps \( {U}_{1} \times {a}_{2} \) into \( {V}_{1} ... | Indeed, it is obtained as a composite map\n\n\[ \n{U}_{1} \rightarrow {U}_{1} \times {U}_{2} \rightarrow {V}_{1} \times {V}_{2} \rightarrow {V}_{1} \n\]\n\nthe first map being an inclusion and the third a projection. | Yes |
Proposition 2.2. Let \( X, Y \) be manifolds of class \( {C}^{p}\left( {p \geqq 1}\right) \) modeled on Banach spaces. Let \( f : X \rightarrow Y \) be a \( {C}^{p} \) -morphism. Let \( x \in X \) . Then:\n\n(i) \( f \) is an immersion at \( x \) if and only if there exists a chart \( \left( {U,\varphi }\right) \) at \... | Proof. This is an immediate consequence of Corollaries 5.4 and 5.6 of the inverse mapping theorem. | Yes |
Proposition 2.4. Let \( X, Y \) be manifolds of class \( {C}^{p}\left( {p \geqq 1}\right) \) modeled on Banach spaces. Let \( f : X \rightarrow Y \) be a \( {C}^{p} \) -morphism, and \( W \) a submanifold of \( Y \) . The map \( f \) is transversal over \( W \) if and only if for each \( x \in X \) such that \( f\left(... | Proof. If \( f \) is transversal over \( W \), then for each point \( x \in X \) such that \( f\left( x\right) \) lies in \( W \), we choose charts as in the definition, and reduce the question to one of maps of open subsets of Banach spaces. In that case, the conclusion concerning the tangent spaces follows at once fr... | Yes |
Proposition 2.5. Let \( f : X \rightarrow Z \) and \( g : Y \rightarrow Z \) be two \( {C}^{p} \) -morphisms with \( p \geqq 1 \) . If they are transversal, then\n\n\[{\left( f \times g\right) }^{-1}\left( {\Delta }_{Z}\right)\]\n\ntogether with the natural morphisms into \( X \) and \( Y \) (obtained from the projecti... | Proof. Obvious. | No |
Proposition 2.6. Assume that each \( {P}_{i} \) admits a manifold structure (compatible with its topology) such that these maps are morphisms, making \( {P}_{i} \) into a fiber product of \( {f}_{i} \) and \( {g}_{i} \) . Then \( P \), with its natural projections, is a fiber product of \( f \) and \( g \) . | To prove the above assertion, we observe that the \( {P}_{i} \) form a covering of \( P \) . Furthermore, the manifold structure on \( {P}_{i} \cap {P}_{j} \) induced by that of \( {P}_{i} \) or \( {P}_{j} \) must be the same, because it is the unique fiber product structure over \( {V}_{i} \cap {V}_{j} \), for the map... | Yes |
Proposition 3.1. If \( X \) is a paracompact space, and if \( \left\{ {U}_{i}\right\} \) is an open covering, then there exists a locally finite open covering \( \left\{ {V}_{i}\right\} \) such that \( {V}_{i} \subset {U}_{i} \) for each \( i \) . | Proof. Let \( \left\{ {V}_{k}\right\} \) be a locally finite open refinement of \( \left\{ {U}_{i}\right\} \) . For each \( k \) there is an index \( i\left( k\right) \) such that \( {V}_{k} \subset {U}_{i\left( k\right) } \) . We let \( {W}_{i} \) be the union of those \( {V}_{k} \) such that \( i\left( k\right) = i \... | Yes |
Proposition 3.2. If \( X \) is paracompact, then \( X \) is normal. If, furthermore, \( \left\{ {U}_{i}\right\} \) is a locally finite open covering of \( X \), then there exists a locally finite open covering \( \left\{ {V}_{i}\right\} \) such that \( {\bar{V}}_{i} \subset {U}_{i} \) . | Proof. We refer the reader to Bourbaki [Bou 68]. | No |
Theorem 3.3. Let \( X \) be a manifold which is locally compact, Hausdorff, and whose topology has a countable base. Given an open covering of \( X \) , then there exists an atlas \( \left\{ \left( {{V}_{k},{\varphi }_{k}}\right) \right\} \) such that the covering \( \left\{ {V}_{k}\right\} \) is locally finite and sub... | Proof. Let \( {U}_{1},{U}_{2},\ldots \) be a basis for the open sets of \( X \) such that each \( {\bar{U}}_{i} \) is compact. We construct inductively a sequence \( {A}_{1},{A}_{2},\ldots \) of compact sets whose union is \( X \), such that \( {A}_{i} \) is contained in the interior of \( {A}_{i + 1} \) . We let \( {A... | Yes |
Corollary 3.4. Let \( X \) be a manifold which is locally compact Hausdorff, and whose topology has a countable base. Then \( X \) admits partitions of unity. | Proof. Let \( \left\{ \left( {{V}_{k},{\varphi }_{k}}\right) \right\} \) be as in the theorem, and \( {W}_{k} = {\varphi }_{k}^{-1}\left( {B}_{1}\right) \) . We can find a function \( {\psi }_{k} \) of class \( {C}^{p} \) such that \( 0 \leqq {\psi }_{k} \leqq 1 \), such that \( {\psi }_{k}\left( x\right) = 1 \) for \(... | Yes |
Lemma 3.6. Let \( E \) be a separable Hilbert space and \( A \subset E \) a closed nonempty subset. Then there exists a real \( {C}^{\infty } \) function \( \psi \) on \( E \) such that \( \psi \left( x\right) = 0 \) for \( x \in A \) and \( \psi \left( x\right) > 0 \) for \( x \notin A \) . | Proof. Let \( h \) be as in Lemma 3.5. Since \( E \) is separable, there exists a sequence \( \left\{ {x}_{n}\right\} \) in the complement \( {A}^{c} \), and dense in this complement. Then\n\n\[ \n{A}^{c} = \mathop{\bigcup }\limits_{n}{B}_{n} \n\]\n\nwhere \( {B}_{n} = B\left( {{x}_{n},{r}_{n}}\right) \) is the ball of... | Yes |
Theorem 3.7. Let \( {A}_{1},{A}_{2} \) be non-void, closed, disjoint subsets of a separable Hilbert space \( \mathbf{E} \) . Then there exists a \( {C}^{\infty } \) -function \( \psi : \mathbf{E} \rightarrow \mathbf{R} \) such that \( \psi \left( x\right) = 0 \) if \( x \in {A}_{1} \) and \( \psi \left( x\right) = 1 \)... | Proof. In the previous lemma, we use functions \( {\psi }_{1} \) and \( {\psi }_{2} \) corresponding to the closed sets \( {A}_{1} \) and \( {A}_{2} \), and we let\n\n\[ \psi = \frac{{\psi }_{1}}{{\psi }_{1} + {\psi }_{2}} \]\n\nto conclude the proof. | No |
Corollary 3.8. Let \( X \) be a paracompact manifold of class \( {C}^{p} \), modeled on a separable Hilbert space \( \mathbf{E} \) . Then \( X \) admits partitions of unity (of class \( {C}^{p} \) ). | Proof. It is trivially verified that an open ball of finite radius in \( \mathbf{E} \) is \( {C}^{\infty } \) -isomorphic to \( \mathbf{E} \) . (We reproduce the formula in Chapter VII.) Given any point \( x \in X \), and a neighborhood \( N \) of \( x \), we can therefore always find a chart \( \left( {G,\gamma }\righ... | Yes |
Proposition 4.1. Let \( f : U \rightarrow \mathbf{F} \) and \( g : U \rightarrow \mathbf{F} \) be two morphisms of class \( {C}^{p}\left( {p \geqq 1}\right) \) defined on an open subset \( U \) of \( \mathbf{E} \) . Assume that \( f \) and \( g \) have the same restriction to \( U \cap {\mathbf{E}}_{\lambda }^{ + } \) ... | Proof. After considering the difference of \( f \) and \( g \), we may assume without loss of generality that the restriction of \( f \) to \( U \cap {\mathbf{E}}_{\lambda }^{ + } \) is 0 . It is then obvious that \( {f}^{\prime }\left( x\right) = 0 \) . | Yes |
Proposition 4.2. Let \( U \) be open in \( \mathbf{E} \) . Let \( \mu \) be a non-zero functional on \( \mathbf{F} \) and let \( f : U \rightarrow {\mathbf{F}}_{\mu }^{ + } \) be a morphism of class \( {C}^{p} \) with \( p \geqq 1 \) . If \( x \) is a point of \( U \) such that \( f\left( x\right) \) lies in \( {\mathb... | Proof. Without loss of generality, we may assume that \( x = 0 \) and \( f\left( x\right) = 0 \) . Let \( W \) be a given neighborhood of 0 in \( \mathbf{F} \) . Suppose that we can find a small element \( v \in \mathbf{E} \) such that \( \mu {f}^{\prime }\left( 0\right) v \neq 0 \) . We can write (for small \( t) \) :... | Yes |
Proposition 4.3. Let \( \lambda \) be a functional on \( \mathbf{E} \) and \( \mu \) a functional on \( \mathbf{F} \) . Let \( U \) be open in \( {\mathbf{E}}_{\lambda }^{ + } \) and \( V \) open in \( {\mathbf{F}}_{\mu }^{ + } \) and assume \( U \cap {\mathbf{E}}_{\lambda }^{0}, V \cap {\mathbf{F}}_{\mu }^{0} \) are n... | Proof. By the functoriality of the derivative, we know that \( {f}^{\prime }\left( x\right) \) is a toplinear isomorphism for each \( x \in U \) . Our first assertion follows from the preceding proposition. We also see that no interior point of \( U \) maps on a boundary point of \( V \) and conversely. Thus \( f \) in... | Yes |
Proposition 1.1. Let \( \mathbf{E},\mathbf{F} \) be finite dimensional vector spaces. Let \( U \) be open in some Banach space. Let\n\n\[ f : U \times \mathbf{E} \rightarrow \mathbf{F} \]\n\nbe a morphism such that for each \( x \in U \), the map\n\n\[ {f}_{x} : \mathbf{E} \rightarrow \mathbf{F} \]\n\ngiven by \( {f}_{... | Proof. We can write \( \mathbf{F} = {\mathbf{R}}_{1} \times \cdots \times {\mathbf{R}}_{n} \) ( \( n \) copies of \( \mathbf{R} \) ). Using the fact that \( L\left( {\mathbf{E},\mathbf{F}}\right) = L\left( {\mathbf{E},{\mathbf{R}}_{1}}\right) \times \cdots \times L\left( {\mathbf{E},{\mathbf{R}}_{n}}\right) \), it will... | Yes |
Proposition 1.2. Let \( X \) be a manifold, and \( \pi : E \rightarrow X \) a mapping from some set \( E \) into \( X \) . Let \( \left\{ {U}_{i}\right\} \) be an open covering of \( X \), and for each \( i \) suppose that we are given a Banach space \( \mathbf{E} \) and a bijection (commuting with the projection on \(... | Proof. By Proposition 3.10 of Chapter I and our condition VB 3, we conclude that the map\n\n\[{\tau }_{j}{\tau }_{i}^{-1} : \left( {{U}_{i} \cap {U}_{j}}\right) \times \mathbf{E} \rightarrow \left( {{U}_{i} \cap {U}_{j}}\right) \times \mathbf{E}\]\n\nis a morphism, and in fact an isomorphism since it has an inverse. Fr... | Yes |
Proposition 1.3. Let \( \pi ,{\pi }^{\prime } \) be two vector bundles over manifolds \( X,{X}^{\prime } \) respectively. Let \( {f}_{0} : X \rightarrow {X}^{\prime } \) be a morphism, and suppose that we are given for each \( x \in X \) a continuous linear map\n\n\[ \n{f}_{x} : {\pi }_{x} \rightarrow {\pi }_{{f}_{0}\l... | Proof. One must first check that \( f \) is a morphism. This can be done under the assumption that \( \pi ,{\pi }^{\prime } \) are trivial, say equal to \( U \times \mathbf{E} \) and \( {U}^{\prime } \times {\mathbf{E}}^{\prime } \) (following the notation of VB Mor 2), with trivialising maps equal to the identity. Our... | Yes |
Proposition 3.1. Let \( X \) be a manifold and let\n\n\[ f : {\pi }^{\prime } \rightarrow \pi \]\n\nbe a VB-morphism of vector bundles over \( X \) . Assume that, for each \( x \in X \), the continuous linear map\n\n\[ {f}_{x} : {E}_{x}^{\prime } \rightarrow {E}_{x} \]\n\nis injective and splits. Then the sequence\n\n\... | Proof. We can assume that \( X \) is connected and that the fibers of \( {E}^{\prime } \) and \( E \) are constant, say equal to the Banach spaces \( {\mathbf{E}}^{\prime } \) and \( \mathbf{E} \) . Let \( a \in X \) . Corresponding to the splitting of \( {f}_{a} \) we know that we have a product decomposition \( \math... | Yes |
Proposition 3.2. Let \( X \) be a manifold and let\n\n\[ g : \pi \rightarrow {\pi }^{\prime \prime } \]\n\nbe a VB-morphism of vector bundles over \( X \) . Assume that for each \( x \in X \), the continuous linear map\n\n\[ {g}_{x} : {E}_{x} \rightarrow {E}_{x}^{\prime \prime } \]\n\n is surjective and has a kernel th... | Proof. It is dual to the preceding one and we leave it to the reader. | No |
Theorem 4.1. Let \( \lambda \) be a functor as above, of class \( {C}^{p}, p \geqq 0 \) . Then for each manifold \( X \), there exists a functor \( {\lambda }_{X} \), on vector bundles (of class \( {C}^{p} \) ) satisfying the following properties. For any bundles \( \alpha ,\beta \) in \( \operatorname{VB}\left( {X,\ma... | Proof. We may assume that \( X \) is connected, so that all the fibers are toplinearly isomorphic to a fixed space. For each open subset \( U \) of \( X \) we let the total space \( {\lambda }_{U}\left( {{E}_{\alpha },{E}_{\beta }}\right) \) of \( {\lambda }_{U}\left( {\alpha ,\beta }\right) \) be the union of the sets... | Yes |
If \( \mu \) is another functor of class \( {C}^{p} \) with the same variance as \( \lambda \), and if we have a natural transformation of functors \( t : \lambda \rightarrow \mu \), then for each \( X \), the mapping\n\n\[ \n{t}_{X} : {\lambda }_{X} \rightarrow {\mu }_{X} \n\]\n\ndefined on each fiber by the map\n\n\[... | Proof. For simplicity of notation, assume that \( \lambda \) and \( \mu \) are both functors of one variable, and both covariant. For each open set \( U = {U}_{i} \) of a trivializing covering for \( \beta \), we have a commutative diagram:  be manifolds and \( {f}_{0} : X \rightarrow Y \) a morphism. Let \( \alpha ,\beta \) be vector bundles over \( X, Y \) respectively, and let \( f, g : \alpha \rightarrow \beta \) be two VB-morphisms over \( {f}_{0} \) . Then the map \( f + g \) defined by the formula\n\n\[ \n{\left( f + ... | Proof. Both assertions are immediate consequences of Proposition 3.10 of Chapter I. | No |
Proposition 5.2. Let \( X \) be a manifold admitting partitions of unity. Let \( 0 \rightarrow \alpha \overset{f}{ \rightarrow }\beta \) be an exact sequence of vector bundles over \( X \) . Then there exists a surjective VB-morphism \( g : \beta \rightarrow \alpha \) whose kernel splits at each point, such that \( g \... | Proof. By the definition of exact sequence, there exists a partition of unity \( \left\{ \left( {{U}_{i},{\psi }_{i}}\right) \right\} \) on \( X \) such that for each \( i \), we can split the sequence over \( {U}_{i} \) . In other words, there exists for each \( i \) a VB-morphism\n\n\[ \n{g}_{i} : \beta \left| {{U}_{... | Yes |
Proposition 5.3. Let \( X \) be a manifold admitting partitions of unity. Let \( \pi \) be a vector bundle of finite type in \( \operatorname{VB}\left( {X,\mathbf{E}}\right) \), where \( \mathbf{E} \) is a Banach space. Then there exists an integer \( k > 0 \) and a vector bundle \( \alpha \) in \( \operatorname{VB}\le... | Proof. We shall prove that there exists an exact sequence\n\n\[ 0 \rightarrow \pi \overset{f}{ \rightarrow }\beta \]\n\nwith \( {E}_{\beta } = X \times {\mathbf{E}}^{k} \) . Our theorem will follow from the preceding proposition.\n\nLet \( \left\{ {{U}_{i},{\tau }_{i})}\right\} \) be a finite trivialization of \( \pi \... | Yes |
Proposition 1.1. Let \( J \) be an open interval of \( \mathbf{R} \) containing 0, and \( U \) open in the Banach space \( \mathbf{E} \) . Let \( {x}_{0} \) be a point of \( U \), and \( a > 0, a < 1 \) a real number such that the closed ball \( {\bar{B}}_{3a}\left( {x}_{0}\right) \) lies in \( U \) . Assume that we ha... | Proof. Let \( {I}_{b} \) be the closed interval \( - b \leqq t \leqq b \), and let \( x \) be a fixed point in \( {\bar{B}}_{a}\left( {x}_{0}\right) \) . Let \( M \) be the set of continuous maps\n\n\[ \n\alpha : {I}_{b} \rightarrow {\bar{B}}_{2a}\left( {x}_{0}\right) \n\]\n\nof the closed interval into the closed ball... | Yes |
Corollary 1.2. The local flow \( \alpha \) in Proposition 1.1 is continuous. Furthermore, the map \( x \mapsto {\alpha }_{x} \) of \( {\bar{B}}_{a}\left( {x}_{0}\right) \) into the space of curves is continuous, and in fact satisfies a Lipschitz condition. | Proof. The second statement obviously implies the first. So fix \( x \) in \( {\bar{B}}_{a}\left( {x}_{0}\right) \) and take \( y \) close to \( x \) in \( {\bar{B}}_{a}\left( {x}_{0}\right) \) . We let \( {S}_{x} \) be the shrinking map of the theorem, corresponding to the initial condition \( x \) . Then\n\n\[ \begin... | Yes |
Theorem 1.3 (Uniqueness Theorem). Let \( U \) be open in \( \mathbf{E} \) and let \( f : U \rightarrow E \) be a vector field of class \( {C}^{p}, p \geqq 1 \) . Let\n\n\[ \n{\alpha }_{1} : {J}_{1} \rightarrow U\;\text{ and }\;{\alpha }_{2} : {J}_{2} \rightarrow U \]\n\nbe two integral curves for \( f \) with the same ... | Proof. Let \( Q \) be the set of numbers \( b \) such that \( {\alpha }_{1}\left( t\right) = {\alpha }_{2}\left( t\right) \) for\n\n\[ \n0 \leqq t < b. \]\n\nThen \( Q \) contains some number \( b > 0 \) by the local uniqueness theorem. If \( Q \) is not bounded from above, the equality of \( {\alpha }_{1}\left( t\righ... | Yes |
Proposition 1.4. Let \( {\varphi }_{1} \) and \( {\varphi }_{2} \) be two \( {\epsilon }_{1} \) - and \( {\epsilon }_{2} \) -approximate solutions of \( f \) on \( {J}_{0} \) respectively, and let \( \epsilon = {\epsilon }_{1} + {\epsilon }_{2} \) . Assume that \( f \) is Lipschitz with constant \( K \) on \( U \) unif... | Proof. By assumption, we have\n\n\[ \left| {{\varphi }_{1}^{\prime }\left( t\right) - f\left( {t,{\varphi }_{1}\left( t\right) }\right) }\right| \leqq {\epsilon }_{1} \]\n\n\[ \left| {{\varphi }_{2}^{\prime }\left( t\right) - f\left( {t,{\varphi }_{2}\left( t\right) }\right) }\right| \leqq {\epsilon }_{2} \]\n\nFrom th... | Yes |
Lemma 1.5. Let \( g \) be a positive real valued function on an interval, bounded by a number \( L \) . Let \( {t}_{0} \) be in the interval, say \( {t}_{0} \leqq t \), and assume that there are numbers \( A, K \geqq 0 \) such that\n\n\[ g\left( t\right) \leqq A + K{\int }_{{t}_{0}}^{t}g\left( u\right) {du} \]\n\nThen ... | Proof. The statement is an assumption for \( n = 1 \) . We proceed by induction. We integrate from \( {t}_{0} \) to \( t \), multiply by \( K \), and use the recurrence relation. The statement with \( n + 1 \) then drops out of the statement with \( n \) . | No |
Corollary 1.6. Let \( f : J \times U \rightarrow \mathbf{E} \) be continuous, and satisfy a Lipschitz condition on \( U \) uniformly with respect to \( J \) . Let \( {x}_{0} \) be a point of \( U \) . Then there exists an open subinterval \( {J}_{0} \) of \( J \) containing 0, and an open subset of \( U \) containing \... | Proof. Given \( x, y \) in \( {U}_{0} \) we let \( {\varphi }_{1}\left( t\right) = \alpha \left( {t, x}\right) \) and \( {\varphi }_{2}\left( t\right) = \alpha \left( {t, y}\right) \), using Proposition 1.6 to get \( {J}_{0} \) and \( {U}_{0} \) . Then \( {\epsilon }_{1} = {\epsilon }_{2} = 0 \) . For \( s, t \) in \( ... | Yes |
Corollary 1.7. Let \( J \) be an open interval of \( \mathbf{R} \) containing 0 and let \( U \) be open in \( \mathbf{E} \). Let \( f : J \times U \rightarrow \mathbf{E} \) be a continuous map, which is Lipschitz on \( U \) uniformly for every compact subinterval of \( J \). Let \( {t}_{0} \in J \) and let \( {\varphi ... | Proof. We can take \( \epsilon = 0 \) in the proposition. | Yes |
Corollary 1.8. Let \( J \) be the open interval \( \left( {a, b}\right) \) and let \( U \) be open in \( \mathbf{E} \) . Let \( f : J \times U \rightarrow \mathbf{E} \) be a continuous map, which is Lipschitz on \( U \), uniformly for every compact subset of \( J \). Let \( \alpha \) be an integral curve of \( f \), de... | Proof. From the integral expression for \( \alpha \), namely \[ \alpha \left( t\right) = \alpha \left( {t}_{0}\right) + {\int }_{{t}_{0}}^{t}f\left( {u,\alpha \left( u\right) }\right) {du} \] we see that for \( {t}_{1},{t}_{2} \) in \( \left( {{b}_{0} - \epsilon ,{b}_{0}}\right) \) we have \[ \left| {\alpha \left( {t}_... | Yes |
Corollary 1.10. Let the notation be as in Proposition 1.9. For each \( x \in V \) and \( z \in E \) the curve\n\n\[ \beta \left( {t, x, z}\right) = \lambda \left( {t, x}\right) z \]\n\nwith initial condition \( \beta \left( {0, x, z}\right) = z \) is a solution of the differential\nequation\n\n\[ {D}_{1}\beta \left( {t... | Proof. Obvious. | No |
Lemma 1.13. Let \( {x}_{0} \in U \) . Let \( a > 0 \) be such that \( {Df} \) is bounded, say by a number \( {C}_{1} > 0 \), on the ball \( {B}_{a}\left( {x}_{0}\right) \) (we can always find such a since \( {Df} \) is continuous at \( \left. {x}_{0}\right) \) . Let \( b < 1/{C}_{1} \) . Then \( {D}_{2}T\left( {x,\sigm... | Proof. We have an estimate\n\n\[ \left| {{\int }_{0}^{t}{Df}\left( {\sigma \left( u\right) }\right) h\left( u\right) {du}}\right| \leqq b{C}_{1}\parallel h\parallel .\n\]\n\nThis means that\n\n\[ \left| {{D}_{2}T\left( {x,\sigma }\right) + I}\right| < 1 \]\n\nand hence that \( {D}_{2}T\left( {x,\sigma }\right) \) is in... | Yes |
Lemma 1.15. Let \( f : U \rightarrow E \) be a \( {C}^{p} \) vector field on the open set \( U \) of \( E \), and let \( \alpha \) be its flow. Abbreviate \( \alpha \left( {t, x}\right) \) by \( {tx} \), if \( \left( {t, x}\right) \) is in the domain of definition of the flow. Let \( x \in U \) . If \( {t}_{0} \) lies ... | Proof. The two curves defined by\n\n\[ t \mapsto \alpha \left( {t,\alpha \left( {{t}_{0}, x}\right) }\right) \;\text{ and }\;t \mapsto \alpha \left( {t + {t}_{0}, x}\right) \]\n\nare integral curves of the same vector field, with the same initial condition \( {t}_{0}x \) at \( t = 0 \) . Hence they have the same domain... | Yes |
Theorem 2.1. Let \( {\alpha }_{1} : {J}_{1} \rightarrow X \) and \( {\alpha }_{2} : {J}_{2} \rightarrow X \) be two integral curves of the vector field \( \xi \) on \( X \), with the same initial condition \( {x}_{0} \) . Then \( {\alpha }_{1} \) and \( {\alpha }_{2} \) are equal on \( {J}_{1} \cap {J}_{2} \) . | Proof. Let \( {J}^{ * } \) be the set of points \( t \) such that \( {\alpha }_{1}\left( t\right) = {\alpha }_{2}\left( t\right) \) . Then \( {J}^{ * } \) certainly contains a neighborhood of 0 by the local uniqueness theorem. Furthermore, since \( X \) is Hausdorff, we see that \( {J}^{ * } \) is closed. We must show ... | Yes |
Theorem 2.2. Let \( \xi \) be a vector field on \( X \), and \( \alpha \) its flows. Let \( x \) be a point of \( X \) . If \( {t}_{0} \) lies in \( J\left( x\right) \), then\n\n\[ J\left( {{t}_{0}x}\right) = J\left( x\right) - {t}_{0} \]\n\n(translation of \( J\left( x\right) \) by \( - {t}_{0} \) ), and we have for a... | Proof. Our first assertion follows immediately from the maximality assumption concerning the domains of the integral curves. The second is equivalent to saying that the two curves given by the left-hand side and right-hand side of the last equality are equal. They are both integral curves for the vector field, with ini... | Yes |
Theorem 2.3. Let \( \xi \) be a vector field on \( X \), and \( x \) a point of \( X \) . Assume that \( {t}^{ + }\left( x\right) < \infty \) . Given a compact set \( A \subset X \), there exists \( \epsilon > 0 \) such that for all \( t > {t}^{ + }\left( x\right) - \epsilon \), the point \( {tx} \) does not lie in \( ... | Proof. Suppose such \( \epsilon \) does not exist. Then we can find a sequence \( {t}_{n} \) of real numbers approaching \( {t}^{ + }\left( x\right) \) from below, such that \( {t}_{n}x \) lies in \( A \) . Since \( A \) is compact, taking a subsequence if necessary, we may assume that \( {t}_{n}x \) converges to a poi... | Yes |
Proposition 2.5. Let \( \\mathbf{E} \) be a Banach space, and \( X \) an \( \\mathbf{E} \) -manifold. Let \( \\xi \) be a vector field on \( X \) . Assume that there exist numbers \( a > 0 \) and \( K > 0 \) such that every point \( x \) of \( X \) admits a chart \( \\left( {U,\\varphi }\\right) \) at \( x \) such that... | Proof. This follows at once from the global continuation theorem, and the uniformity of Proposition 1.1. | No |
Theorem 2.9. Let \( \xi \) be a vector field on \( X \) and \( \alpha \) its flow. Let \( {\mathfrak{D}}_{t}\left( \xi \right) \) be the set of points \( x \) of \( X \) such that \( \left( {t, x}\right) \) lies in \( \mathfrak{D}\left( \xi \right) \) . Then \( {\mathfrak{D}}_{t}\left( \xi \right) \) is open for each \... | Proof. Immediate from the preceding theorem. | No |
Theorem 2.9. Let \( \xi \) be a vector field on \( X \) and \( \alpha \) its flow. Let \( {\mathfrak{D}}_{t}\left( \xi \right) \) be the set of points \( x \) of \( X \) such that \( \left( {t, x}\right) \) lies in \( \mathfrak{D}\left( \xi \right) \) . Then \( {\mathfrak{D}}_{t}\left( \xi \right) \) is open for each \... | Proof. Immediate from the preceding theorem. | No |
Proposition 2.11. If \( \alpha \) is an integral curve of a \( {C}^{1} \) vector field, \( \xi \), and \( \alpha \) passes through a critical point, then \( \alpha \) is constant, that is \( \alpha \left( t\right) = {x}_{0} \) for all \( t \) . | Proof. The constant curve through \( {x}_{0} \) is an integral curve for the vector field, and the uniqueness theorem shows that it is the only one. | No |
Proposition 2.12. Let \( \xi \) be a vector field and \( \alpha \) an integral curve for \( \xi \) . Assume that all \( t \geqq 0 \) are in the domain of \( \alpha \), and that\n\n\[ \mathop{\lim }\limits_{{t \rightarrow 0}}\alpha \left( t\right) = {x}_{1} \]\n\nexists. Then \( {x}_{1} \) is a critical point for \( \xi... | Proof. Selecting \( t \) large, we may assume that we are dealing with the local representation \( f \) of the vector field near \( {x}_{1} \) . Then for \( {t}^{\prime } > t \) large, we have\n\n\[ \alpha \left( {t}^{\prime }\right) - \alpha \left( t\right) = {\int }_{t}^{{t}^{\prime }}f\left( {\alpha \left( u\right) ... | Yes |
Proposition 2.13. Suppose on the other hand that \( {x}_{0} \) is not a critical point of the vector field \( \xi \) . Then there exists a chart at \( {x}_{0} \) such that the local representation of the vector field on this chart is constant. | Proof. In an arbitrary chart the vector field has a representation as a morphism\n\n\[\n\xi : U \rightarrow E\n\]\n\nnear \( {x}_{0} \) . Let \( \alpha \) be its flow. We wish to \ | No |
Proposition 3.2. In a chart \( U \times \mathbf{E} \) for \( {TX} \), let \( f : U \times \mathbf{E} \rightarrow \mathbf{E} \times \mathbf{E} \) represent \( F \), with \( f = \left( {{f}_{1},{f}_{2}}\right) \) . Then \( f \) represents a spray if and only if, for all \( s \in \mathbf{R} \) we have\n\n\[ \n{f}_{2}\left... | Proof. The proof follows at once from the definitions and the formula giving the chart representation of \( s{\left( {s}_{TX}\right) }_{ * } \) .\n\nThus we see that the condition SPR 1 (in addition to being a second-order vector field), simply means that \( {f}_{2} \) is homogeneous of degree 2 in the variable \( v \)... | Yes |
Proposition 3.4. Suppose we are given a covering of the manifold \( X \) by open sets corresponding to charts \( U, V,\ldots \), and for each \( U \) we are given a morphism \[ {B}_{U} : U \rightarrow {L}_{\mathrm{{sym}}}^{2}\left( {\mathbf{E},\mathbf{E}}\right) \] which transforms according to the formula of Propositi... | Proof. We leave the verification to the reader. | No |
Theorem 4.1. Let \( X \) be a manifold and \( F \) a spray on \( X \) . Then\n\n\[{\exp }_{x} : {T}_{x} \rightarrow X\]\n\ninduces a local isomorphism at \( {0}_{x} \), and in fact \( {\left( {\exp }_{x}\right) }_{ * } \) is the identity at \( {0}_{x} \) . | Proof. We prove the second assertion first because the main assertion follows from it by the inverse mapping theorem. Furthermore, since \( {T}_{x} \) is a vector space, it suffices to determine the derivative of \( {\exp }_{x} \) on rays, in other words, to determine the derivative with respect to \( t \) of a curve \... | Yes |
Proposition 4.2. The images of straight segments through the origin in \( {T}_{x} \), under the exponential map \( {\exp }_{x} \), are geodesics. In other words, if \( v \in {T}_{x} \) and we let\n\n\[ \n\alpha \left( {v, t}\right) = {\alpha }_{v}\left( t\right) = {\exp }_{x}\left( {tv}\right) \n\]\n\nthen \( {\alpha }... | Proof. The first statement by definition means that \( {\alpha }_{v}^{\prime } \) is an integral curve of the spray \( F \) . Indeed, by the SPR conditions, we know that\n\n\[ \n\alpha \left( {v, t}\right) = {\alpha }_{v}\left( t\right) = \pi {\beta }_{tv}\left( 1\right) = \pi {\beta }_{v}\left( t\right) , \n\]\n\nand ... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.