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Example 2. (Spaces of integrable functions). Let \( \left( {\Omega ,\sum ,\mu }\right) \) be a positive measure space and let \( p \geq 1 \) . The space \( {L}^{p} \equiv {L}^{p}\left( {\Omega ,\mu ,\mathbb{F}}\right) \) is the linear space of all (equivalence classes of) \( p \) -th power \( \mu \) -integrable functio... | \[ \parallel f{\parallel }_{p} \equiv {\left( {\int }_{\Omega }{\left| f\left( \cdot \right) \right| }^{p}d\mu \right) }^{1/p}. \] | Yes |
Example 3. (Spaces of measures). Let \( \left( {\Omega ,\sum }\right) \) be a measurable space. Then the space \( \mathcal{M}\left( {\Omega ,\sum ,\mathbb{F}}\right) \) is the linear space of all \( \mathbb{F} \) -valued countably additive set functions \( \mu \) defined on the \( \sigma \) -algebra \( \sum \), normed ... | To say that \( \left\{ {A}_{i}\right\} \) partitions \( \Omega \) means that \( \left\{ {A}_{i}\right\} \) is a sequence of pairwise disjoint subsets of \( \sum \) whose union is \( \Omega \) . As is well known, if in the right hand side of (10.3) we replace \( \Omega \) by an arbitrary set \( A \in \sum \), then the f... | Yes |
Example 4. (Spaces of Lipschitz functions). Let \( \left( {\Omega, d}\right) \) be a metric space. Then the space \( \operatorname{Lip}\left( {\Omega, d,\mathbb{F}}\right) \) is the linear space of all bounded \( \mathbb{F} \) -valued functions on \( \Omega \) which satisfy a Lipschitz condition on \( \Omega \) (in the... | Convergence in the associated metric is much stronger than uniform convergence; for an illustration, see exercise 2.9. | No |
Let \( X \) be a locally convex Hausdorff space. Then the space \( {X}^{ * } \) separates the points of \( X \) . | Proof. Let \( x \) and \( y \) be distinct points in \( X \) . We are to find a continuous linear functional \( \phi \) on \( X \) such that \( \phi \left( x\right) \neq \phi \left( y\right) \) . Letting \( z = x - y \neq \theta \), we shall find such a \( \phi \) for which \( \phi \left( z\right) \neq 0 \) . Let \( V ... | Yes |
Corollary 2. A closed solid convex subset of a real linear topological space is supported at every boundary point by a closed hyperplane. | This is an immediate consequence of the theorem, and is the topological version of the support theorem of \( \mathbf{{6C}} \) . | No |
Corollary 1. Let \( A \) be a convex subset of a real locally convex space \( X \) . Then \( A \) is closed if and only if it is weakly closed. | Proof. Assume that \( A \) is closed and that \( x \in X \smallsetminus A \) . By \( {11}\mathbf{F}x \) can be strongly separated from \( A \) by a hyperplane \( \left\lbrack {\phi ;\alpha }\right\rbrack \) for some \( \phi \in {X}^{ * } \) . Thus, by part b) of the lemma again, no net in \( {A}_{1} \) can converge wea... | No |
Corollary 2. Let \( X \) be a normed linear space. Then the norm on \( X \) is a weakly lower semicontinuous function. | This result follows directly from the weak closure of the unit ball (and its positive multiples) in \( X \) . The implication is that\n\n(12.1)\n\n\[ \parallel x\parallel \leq \mathop{\lim }\limits_{\delta }\inf \begin{Vmatrix}{x}_{\delta }\end{Vmatrix} \]\n\nwhenever \( \left\{ {{x}_{\delta } : \delta \in D}\right\} \... | Yes |
Corollary 2. Let \( A \) be a subset of a real locally convex space \( X \) . Then \( \operatorname{span}\left( A\right) \) is weakly dense in \( X \) if and only if \( \phi = \theta \) is the only functional in \( {X}^{ * } \) to annul every point of \( A \) . | Proof. Let \( M = \operatorname{span}\left( A\right) \) and let \( {\bar{M}}^{w} \) be its weak closure in \( X \) . Then \( {\bar{M}}^{w} = X \) if and only if \( {\left( {\bar{M}}^{w}\right) }^{ \circ } = {X}^{ \circ } = \{ \theta \} \), since \( {\bar{M}}^{w} \) and \( X \) are each equal to their own bipolars. But ... | Yes |
Corollary 2. Let \( {A}_{1},\ldots ,{A}_{n} \) be closed convex \( \theta \) -neighborhoods in a locally convex space. Then\n\n(12.4)\n\n\[{\left( {A}_{1} \cap \cdots \cap {A}_{n}\right) }^{ \circ } = \operatorname{co}\left( {{A}_{1}^{ \circ } \cap \cdots \cap {A}_{n}^{ \circ }}\right) . | Proof. This is a consequence of the general formula for the polar of the intersection of an arbitrary family of closed convex sets containing \( \theta \) (exercise 2.28) and the Alaoglu-Bourbaki theorem which guarantees that each of the polars \( {A}_{i}^{ \circ } \) is weak*-compact. It remains only to apply exercise... | No |
Corollary 1. Let \( X \) be a separable locally convex space and let \( A \) be a weak*-closed equicontinuous subset of \( {X}^{ * } \) . Then in its (relative) weak* topology \( A \) is a compact metric space. | Proof. This is a consequence of the Alaoglu-Bourbaki theorem (12D) and the preceding result where we choose the set \( G \) to be any countable dense subset of \( X \) . | No |
Corollary 2. Let \( X \) be a normed linear space. Then the following statements are equivalent.\n\na) \( U\left( {X}^{ * }\right) \) is weak*-metrizable;\n\nb) every ball in \( {X}^{ * } \) is weak*-metrizable;\n\nc) \( X \) is separable. | Proof. We need only check that a) implies c). Since every metric space is first countable there exists a countable \( \theta \) -neighborhood base \( \left\{ {V}_{n}\right\} \) in \( U\left( {X}^{ * }\right) \) and hence a sequence \( \left\{ {A}_{n}\right\} \) of finite subsets of \( X \) such that\n\n\[ \left\{ {\phi... | No |
Corollary 2. A lower semicontinuous quasi-concave function fon \( A \) attains its minimum on \( A \) at an extreme point of \( A \) . | Proof. A quasi-concave function is by definition the negative of a quasi-convex function (exercise 1.10). It follows that the sets \( \{ x \in A : \alpha < f\left( x\right) \} \) are open and convex for every real \( \alpha \) . Thus we see that the set \( B \subset A \) where \( f \) attains its minimum satisfies the ... | No |
For every normed linear space \( X \) the extreme points of \( U\left( {X}^{ * }\right) \) separate the points of \( X \) . | We can paraphrase this corollary by stating that to any pair \( x, y \) of distinct points of \( X \) there corresponds a \( \phi \in \operatorname{ext}\left( {U\left( {X}^{ * }\right) }\right) \) such that \( \phi \left( x\right) \neq \phi \left( y\right) \) . The fact that the balls in the dual space of a normed line... | No |
Lemma 1. A (non-zero) cone \( C \) in a real locally convex space \( X \) is locally compact if and only if it has a compact base, in which case \( C \) is necessarily closed. | Proof. Assume that \( C \) is locally compact. There is then a closed convex \( \theta \) -neighborhood \( U \subset X \) for which \( C \cap U \) is compact. Let \( D \) be the intersection of \( C \) with the boundary of \( U \) . Then \( \overline{\mathrm{{co}}}\left( D\right) \subset C \cap U \) and hence is compac... | Yes |
Lemma 2. A (non-empty) closed, convex, locally compact, and line-free set \( A \) in \( X \) has an extreme point. | Proof. We may assume that \( A \) is not compact. Then \( {C}_{A} \) is a non-trivial closed cone in \( X \) (closure follows from equation (8.5)). Further \( {C}_{A} \) is itself locally compact since a translate of it lies in \( A \) . Let \( \phi \in {X}^{ * } \) be a strictly positive linear functional and let \( K... | Yes |
Let \( \Omega \) be the unit disc \( \{ z \in \mathbb{C} : \left| z\right| \leq 1\} \) and let \( X = A\left( \Omega \right) \). As in Example 1, any function in \( X \) which is of modulus one on the boundary \( \partial \Omega \) of \( \Omega \) (the unit circle) is an extreme point of \( U\left( X\right) \). More ge... | (13.8)\n\n\[{\int }_{-\pi }^{\pi }\log \left( {1 - \left| {f\left( {e}^{it}\right) }\right| }\right) {dt} = - \infty .\]\n\n(To see that this condition is necessary, assume that it fails. Select a continuous function \( h \) on \( \partial \Omega \) such that \( \theta \leq h\left( \cdot \right) \leq 1 - \left| {f\left... | Yes |
Let \( X = \operatorname{Lip}\left( {\left\lbrack {0,1}\right\rbrack, d,\mathbb{F}}\right) \) where \( d \) is the usual metric on \( \left\lbrack {0,1}\right\rbrack \) and, as usual, \( \mathbb{F} = \mathbb{R} \) or \( \mathbb{C} \) . Any function \( f \in X \) is differentiable almost everywhere and, in fact, \( {\be... | (To see that this condition is necessary, we can proceed by contradiction. Let \( E \) be a compact subset of \( \{ t \in \left\lbrack {0,1}\right\rbrack : \left| {f\left( t\right) }\right| < 1\} \) with positive measure such that \( {\begin{Vmatrix}{f}^{\prime }\left| E\right| \end{Vmatrix}}_{\infty } < 1 \) . Then we... | Yes |
We consider again \( X = {C}_{b}\left( {\Omega ,\mathbb{F}}\right) \) but now we try to identify \( \operatorname{ext}\left( {U\left( {X}^{ * }\right) }\right) \). In this example we shall assume that \( \Omega \) is a compact Hausdorff space. This is a very important case in practice. Our task is facilitated by the re... | \[ E \equiv \left\{ {{\left. \alpha {\delta }_{i};\right| }_{1}\left| \alpha \right| = 1, t \in \Omega }\right\} \subset \operatorname{ext}\left( {U\left( {X}^{ * }\right) }\right) . \] Now evidently \( {}^{ \circ }E = U\left( X\right) \), so by the bipolar theorem (12C) we have \[ U\left( {X}^{ * }\right) = {}^{ \circ... | Yes |
A normed linear space \( \left( {X,\parallel \cdot \parallel }\right) \) is strictly normed if \( \parallel x + y\parallel = \parallel x\parallel + \parallel y\parallel \) implies that \( x = {ty} \) for some \( t \geq 0 \) or else \( y = \theta \). | This constraint on the norm is easily seen to be equivalent to the geometric condition that \( U\left( X\right) \) be rotund, where a convex set is rotund if every bounding point is an extreme point. From our present point of view such spaces are not very interesting since, for example, condition (13.7) is automaticall... | No |
Then we claim that the identity map \( I \) (where \( I\left( x\right) \equiv x \) for all \( x \in X \) ) is an extreme point of \( U\left( {B\left( X\right) }\right) \) (whether or not \( X \) is strictly normed). | In fact, one can prove the much stronger assertion that \( I \) is a vertex of \( U\left( {B\left( X\right) }\right) \) in the sense that the set \( \left\{ {\phi \in B{\left( X\right) }^{ * } : \parallel \phi \parallel = 1 = \phi \left( I\right) }\right\} \) is total over \( B\left( X\right) \) . In other words, the i... | Yes |
Corollary 2. Let \( f, g \in \operatorname{Conv}\left( A\right) \) and assume both are continuous at \( p \in A \) . Then for any non-negative number’s \( s \) and \( t \) we have\n\n\[ \partial \left( {{sf} + {tg}}\right) \left( p\right) = s\partial f\left( p\right) + t\partial g\left( p\right) \] | Proof. Let \( h = {sf} + {tq} \) . Then\n\n\[ \max \{ \psi \left( x\right) : \psi \in \partial h\left( p\right) \} = {h}^{\prime }\left( {p;x}\right) = s{f}^{\prime }\left( {p;x}\right) + t{g}^{\prime }\left( {p;x}\right) \]\n\n\[ = \overline{\max \left\{ {\psi \left( x\right) :\psi \in \partial f\left( p\right) }\righ... | No |
Consider the special case where \( A = {x}_{o} + M \) is an affine subspace (1C) of \( X \) . For any \( p \in A \) the necessary and sufficient condition that \( p \) solve the program \( \left( {A, f}\right) \) is that \( \partial f\left( p\right) \cap {M}^{ \circ } \neq \varnothing \), where \( {M}^{ \circ } \) is t... | This is because \( F\left( {p;A}\right) = M \) in this case. | No |
Let \( A \) be a convex subset of the (real) linear topological space \( X \) and let \( f \in \operatorname{Conv}\left( A\right) \) . Let \( S : A \rightarrow Y \) be a convex mapping with values in an ordered linear topological space \( Y \) . Assume that the associated system (14.12) is consistent and that \( f\left... | Proof. This is a direct consequence of Corollary 1 and the theorem if we take \( Z = {\mathbb{R}}^{1} \) with the usual ordering and let \( T = f \) . | No |
Lemma 1. Let \( A \) be a closed locally compact convex subset of a strictly normed linear space. For each \( x \in X \) there is a unique best approximation \( {P}_{A}\left( x\right) \) to \( x \) in \( A \) and the map \( {P}_{A} \) is continuous on \( X \) . | Proof. The sets \( A \cap \left( {x + {\lambda U}\left( X\right) }\right) \) are closed, convex, locally compact and non-empty for sufficiently large \( \lambda > 0 \) . Since they have a trivial recession cone, they must also be compact (13C). Their intersection, taken over those \( \lambda \) yielding non-empty sets,... | Yes |
Lemma 2. Let \( X \) be a separable normed linear space.\n\na) There is an equivalent strict norm on \( X \) . | a) By 12F there is a weak*-dense sequence \( \left\{ {\phi }_{n}\right\} \) in \( U\left( {X}^{ * }\right) \) . Define\n\n\[ \sigma \left( x\right) = \parallel x\parallel + {\left( \mathop{\sum }\limits_{{n = 1}}^{\infty }{2}^{-n}{\left| {\phi }_{n}\left( x\right) \right| }^{2}\right) }^{1/2},\;x \in X, \]\n\nwhere \( ... | Yes |
Theorem. Let \( A \) be a closed convex subset of a normed linear space \( X \) , and let \( f \) be a continuous map of \( A \) into a compact subset of \( A \) . Then \( f \) has a fixed point in \( A \) . | Proof. We first reduce the problem to the case where \( A \) is bounded and \( X \) is a separable strictly normed space. This can be done noting that if \( f\left( A\right) \subset K \), a compact set, then we can replace \( A \) by \( B \equiv \overline{\operatorname{co}}\left( K\right) \subset A \) and try to prove ... | Yes |
Lemma 3. The Banach space \( {C}_{b}\left( {\Omega ,\mathbb{F}}\right) \) is separable if and only if the metric space \( \left( {\Omega, d}\right) \) is compact. | Proof. From exercise 1.40 we see that \( {C}_{b}\left( {\Omega ,\mathbb{C}}\right) \) is separable if and only if \( {C}_{b}\left( {\Omega ,\mathbb{R}}\right) \) is, so we just work with the latter space. Suppose that \( \Omega \) is compact. Then \( \Omega \) is 2nd-countable and there is a countable base \( \left\{ {... | No |
Lemma 1. If \( M \) is thin then \( K \subset \operatorname{ext}\left( K\right) + {M}^{ \bot } \) . | Proof. Let \( \phi \in K \) . Then the set \( A \equiv K \cap \left( {\phi + M}\right) \) is compact and so by 13A there exists \( \psi \in \operatorname{ext}\left( A\right) \) . We claim that \( \psi \) is a characteristic function and hence in \( \operatorname{ext}\left( K\right) \) . If not, there exist \( \varepsil... | Yes |
Lemma 2. If \( v \) is non-atomic then any finite set \( M \subset {L}^{1}\left( v\right) \) is thin. | Proof. Suppose that \( M = \left\{ {{f}_{1},\ldots ,{f}_{n}}\right\} \) and that \( E \in \sum \) . Since \( v \) is nonatomic we can partition \( E \) into disjoint sets \( {E}_{1},\ldots ,{E}_{n + 1} \) of positive measure. Let \( A \) be the \( n \times \left( {n + 1}\right) \) matrix with entries \( {\int }_{{E}_{j... | Yes |
Lemma 3. a) \( \operatorname{ext}\left( K\right) \) is weak*-dense in \( K \) if (and only if) \( v \) is non-atomic. | Proof. a) Let \( N \equiv \left\{ {\phi \in {L}^{\infty }\left( v\right) : \left| {{\int }_{\Omega }{f}_{i}{\phi dv}}\right| \leq 1,1 \leq i \leq n}\right\} \) be a weak*- \( \theta \) -neighborhood in \( {L}^{\infty }\left( v\right) \) and set \( \ddot{M} = \left\{ {{f}_{1},\ldots ,{f}_{n}}\right\} \) . Then by Lemmas... | No |
Corollary 1. Let \( T \) be a continuous algebraic isomorphism between Banach spaces \( X \) and \( Y \) . Then \( T \) is a topological isomorphism. | Proof. It must be shown that the inverse mapping \( {T}^{-1} \) is continuous. By hypothesis \( {T}^{-1} \) is defined on all of \( Y \) and is closed on \( Y \) by part b) of the preceding lemma. Therefore, \( {T}^{-1} \) is continuous by the closed graph theorem. | Yes |
Corollary 2. Let \( X \) be a Banach space. A convex subset \( C \) of \( {X}^{ * } \) is weak*-closed if and only if it is bw*-closed. | Proof. Suppose that \( \phi \) does not belong to the bw*-closure of \( C \) . Since the bw*-topology is locally convex we can apply the strong separation theorem and separate \( C \) from \( \phi \) by a bw*-closed hyperplane \( H \) . By the theorem \( H \) is weak*-closed so that \( \phi \) cannot belong to the weak... | Yes |
Lemma 3. Let \( \sigma \) be a sublinear function on a linear space \( X, K \) a convex subset of \( X, u \) a point in \( X \), and \( \alpha ,\beta \), and \( {\beta }^{\prime } \) three positive numbers. If \( \inf \{ \sigma \left( {u + {\beta x}}\right) : x \in K\} > {\alpha \beta } + \sigma \left( u\right) \), the... | Proof. There is a \( \delta > 0 \) such that \( - \sigma \left( u\right) = {\alpha \beta } - \inf \{ \sigma \left( {u + {\beta x}}\right) \) : \( x \in K\} + \delta \) . For any \( {x}_{0},{y}_{1} \in K \), put \( z = \left( {\beta {x}_{0} + {\beta }^{\prime }{y}_{1}}\right) /\left( {\beta + {\beta }^{\prime }}\right) ... | Yes |
Lemma 1. Let \( \left\{ {{V}_{\alpha } : \alpha \in I}\right\} \) be a locally finite open covering of a normal space \( \Omega \) . Then there exists a partition of unity \( \left\{ {{p}_{\alpha } : \alpha \in I}\right\} \) which is subordinate to this covering. | The proof is achieved by shrinking the covering \( \left\{ {V}_{\alpha }\right\} \) to obtain a new open covering \( \left\{ {{W}_{\alpha } : \alpha \in I}\right\} \) such that \( {\bar{W}}_{\alpha } \subset {V}_{\alpha },\alpha \in I \), and then using Urysohn’s lemma to obtain continuous functions \( {q}_{\alpha } : ... | Yes |
Lemma 2. Let \( \Omega \) be a paracompact space, \( X \) a normed linear space, and \( F : \Omega \rightarrow {2}^{X} \) a lower semicontinuous carrier whose values are non-empty convex subsets of \( X \) . Then if \( r > 0 \) there exists a continuous map \( f : \Omega \rightarrow X \) such that \( d\left( {f\left( t... | Proof. For each \( x \in X \) let\n\n\[ \n{\mathcal{O}}_{x} = \{ t \in \Omega : d\left( {x, F\left( t\right) }\right) < r\} .\n\]\n\nThese sets \( {\mathcal{O}}_{x} \) are open in \( \Omega \) because of the lower semicontinuity of \( F \) and therefore \( \left\{ {{\mathcal{O}}_{x} : x \in X}\right\} \) is an open cov... | Yes |
Lemma 1. Let \( A \) be a compact convex subset of a locally convex space. If \( A \) is also metrizable then \( \operatorname{ext}\left( A\right) \) is a Baire space. | Proof. Let \( d \) be a metric that defines the topology on \( A \), and define \( {C}_{n} = \left\{ {\frac{1}{2}\left( {x + y}\right) : x, y \in A\text{and}d\left( {x, y}\right) \geq \frac{1}{n}}\right\} \) . Then the sets \( {C}_{n} \) are closed and \( \mathop{\bigcup }\limits_{n}{C}_{n} = A \smallsetminus \operator... | Yes |
Lemma 2. Let \( Y \) be a Banach space for which \( {Y}^{ * } \) is separable, and let \( A \) be a weak \( * \) -compact subset of \( {Y}^{ * } \) . Then the set of all points of continuity of the identity map: \( {A}_{{w}^{ * }} \rightarrow A \) is weak*-dense in \( A \) . | Proof. For each \( \varepsilon > 0 \) let \( {A}_{\varepsilon } \) be the union of all open subsets of (norm) diameter \( \leq \varepsilon \) . Clearly \( {A}_{\varepsilon } \) is open and we shall show that it is dense in \( A \) . Since \( A \) is separable there is a (norm) dense sequence \( \left\{ {\phi }_{n}\righ... | Yes |
Let \( \Omega \) be a non-compact but locally compact Hausdorff space and let \( {C}_{0}\left( {\Omega ,\mathbb{F}}\right) \) be the Banach space of continuous \( \mathbb{F} \) -valued functions on \( \Omega \) that vanish at infinity, with the usual sup norm. According to exercise 2.30 the unit ball of this space has ... | In particular the sequence space \( {c}_{0} \) has this property. Hence so does the sequence space \( c \), where \( c \) consists by definition of all convergent sequences of scalars. | No |
By contrast with Example 1 the space \( {L}^{1} \) is complemented (in fact, constrained) in its second conjugate space. To prove this assertion we must define a norm-one projection from \( {L}^{1 * * } \cong {L}^{\infty * } \) onto \( {J}_{{L}^{1}}\left( {L}^{1}\right) \) . | Given \( \Phi \in {L}^{\infty * } \) let \( \phi = \Phi \mid C\left( {\left\lbrack {0,1}\right\rbrack ,\mathbb{R}}\right) \) . Let \( g \in {NBV}\left( {\left\lbrack {0,1}\right\rbrack ,\mathbb{R}}\right) \) correspond to \( \phi \) according to formula (22.6). Then the derivative \( {g}^{\prime } \) belongs to \( {L}^... | Yes |
Lemma 1. Let \( \left( {\Omega ,\sum ,\mu }\right) \) be a \( \sigma \) -finite measure space. Then \( {L}^{\infty } \equiv {L}^{\infty }\left( {\Omega ,\mu ,\mathbb{R}}\right) \) is boundedly complete. | Proof. Let \( A \) be a subset of \( {L}^{\infty } \) which is bounded above: there exists \( g \in {L}^{\infty } \) such that \( f \leq g \) (that is, \( \theta \leq g - f\left\lbrack \mu \right\rbrack \) ) for all \( f \in A \) . After replacing \( A \) by the suprema of its finite subsets (if necessary) we may suppo... | Yes |
Lemma 2. Let \( {L}^{\infty } \) be as in Lemma 1 and suppose that \( {L}^{\infty } \) is (congruent with) a subspace of a Banach space \( X \) . Then \( {L}^{\infty } \) is constrained in \( X \) . | Proof. Let \( e \) be the identically one function in \( {L}^{\infty } \) and define a sublinear mapping \( g : X \rightarrow {L}^{\infty } \) by \( g\left( x\right) = \parallel x\parallel e \) . Let \( f \) be the identity map on \( {L}^{\infty } \) . Then \( f \leq g \mid M \), and any extension of \( f \) to an oper... | Yes |
Let \( \mathcal{C} \) consist of all separable Banach spaces and their conjugate spaces. Then \( m \) is universal for \( \mathcal{C} \). | Indeed, let \( X \) be a separable Banach space. Then \( U\left( {X}^{ * }\right) \) is a compact metric space in the weak*-topology, and in particular there is a sequence \( \left\{ {\phi }_{n}\right\} \) that is weak*-dense in \( U\left( {X}^{ * }\right) \) . Define \( T : X \rightarrow m \) by \( T\left( x\right) = ... | Yes |
Let \( X \) be a Banach space. Then there exists a compact Hausdorff space \( \Omega \) with \( \operatorname{dens}\left( \Omega \right) \leq \operatorname{dens}\left( X\right) \) such that \( X \) is congruent to a subspace of \( C\left( {\Omega ,\mathbb{F}}\right) \) . | To see this we let \( \Omega = U\left( {X}^{ * }\right) \) given the (relative) weak*- topology. The map \( x \mapsto \widehat{x} \mid \Omega, x \in X \), is clearly a linear isometry of \( X \) into \( C\left( {\Omega ,\mathbb{F}}\right) \), since\n\n\[ \parallel x\parallel = \sup \{ \left| {\phi \left( x\right) }\rig... | Yes |
A real Banach space \( X \) is congruent with a subspace of \( C \equiv \) \( C\left( {\Omega ,\mathbb{R}}\right) \), where \( \Omega \) is a dispersed compact Hausdorff space,(if and) only if \( \operatorname{ext}{\left( U\left( {X}^{ * }\right) \right) }^{ * } \) is dispersed. | Proof. Suppose that \( T \) is a congruence of \( X \) with a subspace of \( C \) . Let \( \phi \in \operatorname{ext}\left( {U\left( {X}^{ * }\right) }\right) \) . Then \( \phi \circ {T}^{-1} \in \operatorname{ext}\left( {U\left( {T{\left( X\right) }^{ * }}\right) }\right) \) and so extends to a functional \( \psi \in... | No |
Lemma 2. Let \( \Omega \) be a compact metric space which is not dispersed. Then there exists a continuous mapping from \( \Omega \) onto \( \left\lbrack {0,1}\right\rbrack \) . | Proof. Let \( P \) be a perfect subset of \( \Omega \) . We distinguish two cases.\n\na) \( P \) is totally disconnected. In this case \( P \) is known to be homeomorphic to the Cantor set. Since any compact metric space (in particular, \( \left\lbrack {0,1}\right\rbrack \) ) is the continuous image of the Cantor set, ... | No |
Proposition 9.1.2. We have the following properties:\n\n(1) \( {B}_{k}^{\prime }\left( x\right) = k{B}_{k - 1}\left( x\right) \) .\n\n(2) \( {B}_{k}\left( x\right) \) is a monic polynomial of degree \( k \) .\n\n(3) For \( k \neq 1 \) we have \( {B}_{k}\left( 1\right) = {B}_{k}\left( 0\right) = {B}_{k} \), while for \(... | Proof. All these results are immediate consequences of the definition: (1) is equivalent to \( \frac{\partial E\left( {t, x}\right) }{\partial x} = {tE}\left( {t, x}\right) \) ,(2) follows by induction,(3) is equivalent to \( E\left( {t,1}\right) - E\left( {t,0}\right) = t \) ,(4) to the fact that \( E\left( {t,0}\righ... | Yes |
Proposition 9.1.3. We have\n\n\[ \n{B}_{k}\left( {x + 1}\right) = {B}_{k}\left( x\right) + k{x}^{k - 1}, \]\n\n\[ \n{B}_{k}\left( {-x}\right) = {\left( -1\right) }^{k}\left( {{B}_{k}\left( x\right) + k{x}^{k - 1}}\right) , \]\n\n\[ \n{B}_{k}\left( {1 - x}\right) = {\left( -1\right) }^{k}{B}_{k}\left( x\right) , \]\n\n\... | Proof. It is immediate that these formulas are equivalent respectively to the trivial identities \( E\left( {t, x + 1}\right) = E\left( {t, x}\right) + t{e}^{tx}, E\left( {-t, - x}\right) = {e}^{t}E\left( {t, x}\right) = \) \( E\left( {t, x}\right) + t{e}^{tx}, E\left( {-t,1 - x}\right) = E\left( {t, x}\right), E\left(... | Yes |
Proposition 9.1.4. We have the following Taylor series expansions with radii of convergence \( R \) indicated in parentheses:\n\n\[ \operatorname{cotanh}\left( t\right) = \frac{1}{t} + \mathop{\sum }\limits_{{k \geq 1}}{2}^{2k}\frac{{B}_{2k}}{\left( {2k}\right) !}{t}^{{2k} - 1}\;\left( {R = \pi }\right) , \] | Proof. By definition\n\n\[ \operatorname{cotanh}\left( t\right) = \frac{\cosh \left( t\right) }{\sinh \left( t\right) } = \frac{{e}^{t} + {e}^{-t}}{{e}^{t} - {e}^{-t}} = \frac{{e}^{2t} + 1}{{e}^{2t} - 1} = 1 + \frac{1}{t}\frac{2t}{{e}^{2t} - 1}, \] | Yes |
Corollary 9.1.7. The tangent numbers satisfy the recurrence\n\n\\[ \n\\mathop{\\sum }\\limits_{{j = 1}}^{k}\\left( \\begin{array}{l} {2k} - 1 \\\\ {2j} - 1 \\end{array}\\right) {T}_{{2j} - 1} = 1\\text{ for }k > 0,\n\\]\n\nand in particular \\( {T}_{{2k} - 1} \\in \\mathbb{Z} \\) for all \\( k \\geq 1 \\) . | Proof. This immediately follows from the identity \\( \\cosh \\left( t\\right) \\tanh \\left( t\\right) = \\) \\( \\sinh \\left( t\\right) \\), and the details are left to the reader. | No |
Proposition 9.1.9. We have\n\n\[ \n{B}_{2k}\left( {1/4}\right) = {B}_{2k}\left( {3/4}\right) = \frac{{B}_{2k}\left( {1/2}\right) }{{2}^{2k}} = - \frac{1}{{2}^{2k}}\left( {1 - \frac{1}{{2}^{{2k} - 1}}}\right) {B}_{2k}, \n\] | Proof. Multiplying the identity \( 1/\left( {{e}^{t} + 1}\right) = 1/\left( {{e}^{t} - 1}\right) - 2/\left( {{e}^{2t} - 1}\right) \) given above by \( {e}^{t/2} \) and replacing \( t \) by \( {2t} \), we obtain\n\n\[ \n\frac{1}{\cosh \left( t\right) } = \frac{2{e}^{t}}{{e}^{2t} + 1} = - \mathop{\sum }\limits_{{k \geq 1... | Yes |
Corollary 9.1.10. The Euler numbers satisfy the recurrence\n\n\\[ \n\\mathop{\\sum }\\limits_{{j = 0}}^{k}\\left( \\begin{array}{l} {2k} \\\\ {2j} \\end{array}\\right) {E}_{2j} = 0\\text{ for }k > 0, \n\\]\n\nand in particular \\( {E}_{2k} \\in \\mathbb{Z} \\) for all \\( k \\) . | Proof. This immediately follows from the identity \\( \\cosh \\left( t\\right) \\left( {1/\\cosh \\left( t\\right) }\\right) = 1 \\) . It also follows from the second formula of Proposition 9.1.14 below applied to \\( x = y = 1/4 \\) . We thus have \\( {E}_{2k} = - \\mathop{\\sum }\\limits_{{0 \\leq j < k}}\\left( \\be... | Yes |
Proposition 9.1.11. We have\n\n\[ S\\left( {-t, - x}\\right) = - S\\left( {t, x}\\right) - \\frac{1}{{\\left( t - x\\right) }^{2}}, \]\n\n\[ S\\left( {t, x + 1}\\right) = S\\left( {t, x}\\right) + \\frac{1}{{\\left( t - x\\right) }^{2}}, \]\n\n\[ S\\left( {t - y, x}\\right) = S\\left( {t, x + y}\\right) ,\]\n\nand in p... | Proof. Using the formula for \( {B}_{n}\\left( {-x}\\right) \) mentioned above we have\n\n\[ S\\left( {-t, - x}\\right) = \\mathop{\\sum }\\limits_{{k \\geq 0}}{\\left( -1\\right) }^{k + 1}\\frac{{B}_{k}\\left( {-x}\\right) }{{t}^{k + 1}} = - \\mathop{\\sum }\\limits_{{k \\geq 0}}\\frac{{B}_{k}\\left( x\\right) + k{x}^... | Yes |
Proposition 9.1.12. As a formal power series in \( t, S\left( {t, x}\right) \) is the Laplace transform of \( E\left( {t, x}\right) \) ; in other words, we have formally\n\n\[ S\left( {t, x}\right) = {\int }_{0}^{\infty }{e}^{-{tu}}E\left( {u, x}\right) {du}. \]\n\nFurthermore, for \( t > x - 1 \) the above integral co... | Proof. The first statement is clear by expanding \( E\left( {u, x}\right) \) as a power series\nin \( u \) since\n\n\[ {\int }_{0}^{\infty }{e}^{-{tu}}{u}^{k}{du} = \frac{k!}{{t}^{k + 1}} \]\n\nand the second follows since the integrand is continuous everywhere and is asymptotic to \( u{e}^{u\left( {x - 1 - t}\right) }... | Yes |
Corollary 9.1.13. For \( t > x - 1 \) we have\n\n\[ S\left( {t, x}\right) = {\psi }^{\prime }\left( {t - x + 1}\right) = {\psi }^{\prime }\left( {t - x}\right) - \frac{1}{{\left( t - x\right) }^{2}}, \]\n\nwhere \( \psi = {\Gamma }^{\prime }/\Gamma \) is the logarithmic derivative of the gamma function (see Definition ... | Proof. From Corollary 9.6.43 below we have\n\n\[ {\psi }^{\prime }\left( {s + 1}\right) = {\int }_{0}^{\infty }\frac{v{e}^{-{sv}}}{{e}^{v} - 1}{dv} \]\n\nso the result follows from the proposition. | No |
Proposition 9.1.14. For \( k \geq 0 \) we have\n\n\[ \mathop{\sum }\limits_{{j = 0}}^{k}\left( \begin{array}{l} k \\ j \end{array}\right) {y}^{k - j}\frac{{B}_{j + 1}\left( x\right) }{j + 1} = \frac{{B}_{k + 1}\left( {x + y}\right) - {y}^{k + 1}}{k + 1}, \] | Proof. We could give a proof of the first formula directly from the generating function, as we did for Proposition 9.1.3. It is however instructive to give an alternative proof. After all, if we integrate with respect to \( x \) the formula for \( {B}_{k}\left( {x + y}\right) \) given in Proposition 9.1.3 and use \( {B... | Yes |
Corollary 9.1.15. For \( k \geq 0 \) We have\n\n\[ \mathop{\sum }\limits_{{j = 0}}^{{k - 1}}\left( \begin{array}{l} k \\ j \end{array}\right) \frac{{B}_{j + 1}\left( x\right) }{j + 1} = {x}^{k} - \frac{1}{k + 1}, \] | Proof. These formulas are obtained by suitable specializations to \( y = 1 \) or \( y = 1/2 \) of the formulas of the proposition. | No |
Proposition 9.1.16. For any \( k \) and \( m \) in \( {\mathbb{Z}}_{ \geq 0} \) we have\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{{\max \left( {k, m}\right) }}\left( {\left( \begin{array}{l} k \\ j \end{array}\right) + {\left( -1\right) }^{j + 1}\left( \begin{matrix} m \\ j \end{matrix}\right) }\right) \frac{{B}_{k + m + ... | Proof. Consider \( x \) as a fixed parameter and set\n\n\[ {F}_{x}\left( t\right) = \frac{E\left( {t, x}\right) }{t} = \frac{{e}^{tx}}{{e}^{t} - 1} = \frac{1}{t} + \mathop{\sum }\limits_{{k \geq 0}}\frac{{B}_{k + 1}\left( x\right) }{k + 1}\frac{{t}^{k}}{k!}, \]\n\nlet \( D = d/{dt} \) be the differentiation operator wi... | Yes |
Lemma 9.1.17. With the above notation we have\n\n\[ \left( {{e}^{t}{D}^{m}{\left( D + I\right) }^{k} - {D}^{k}{\left( D - I\right) }^{m}}\right) {F}_{x}\left( t\right) = {x}^{k}{\left( x - 1\right) }^{m}{e}^{xt}. \] | Proof. For simplicity write \( {F}_{x} \) instead of \( {F}_{x}\left( t\right) \) . Leibniz’s rule can be written in operator notation\n\n\[ {D}^{N}\left( {{e}^{at}{F}_{x}}\right) = \left( {\mathop{\sum }\limits_{{j = 0}}^{N}\left( \begin{matrix} N \\ j \end{matrix}\right) {a}^{N - j}{e}^{at}{D}^{j}}\right) {F}_{x} = {... | Yes |
Corollary 9.1.18. For any \( k \) and \( m \) in \( {\mathbb{Z}}_{ \geq 0} \) we have\n\n\[ \mathop{\sum }\limits_{{j = 0}}^{{\max \left( {k, m}\right) }}\left( {\left( \begin{array}{l} k \\ j \end{array}\right) + {\left( -1\right) }^{k + m}\left( \begin{matrix} m \\ j \end{matrix}\right) }\right) \frac{{B}_{k + m + 1 ... | Proof. The first four formulas follow by taking \( x = 0 \) in the proposition and using the formulas for the odd Bernoulli numbers. The replacement of \( {\left( -1\right) }^{j} \) by \( \pm {\left( -1\right) }^{k + m} \) and the fact that we begin at \( j = 0 \) removes the special cases. The details are left to the ... | No |
Theorem 9.1.20. (1) For \( n \geq 2 \) even we have\n\n\[ \mathop{\sum }\limits_{{k \geq 1}}\frac{\cos \left( {2\pi kx}\right) }{{k}^{n}} = \frac{{\left( -1\right) }^{n/2 + 1}}{2}\frac{{\left( 2\pi \right) }^{n}{B}_{n}\left( {\{ x\} }\right) }{n!}. \]\n\n(2) For \( n \geq 1 \) odd we have\n\n\[ \mathop{\sum }\limits_{{... | Proof. (1) and (2). Since \( {B}_{n}\left( 1\right) = {B}_{n}\left( 0\right) \) for \( n \neq 1 \), the function \( {B}_{n}\left( {\{ x\} }\right) \) is piecewise \( {C}^{\infty } \) and continuous for \( n \geq 2 \), with simple discontinuities at the integers if \( n = 1 \) . If \( n \geq 2 \) we thus have\n\n\[ {B}_... | Yes |
Corollary 9.1.21. For \( n \geq 1 \) we have\n\n\[ \mathop{\sum }\limits_{{k \geq 1}}\frac{1}{{k}^{2n}} = \frac{{\left( -1\right) }^{n - 1}}{2}\frac{{\left( 2\pi \right) }^{2n}{B}_{2n}}{\left( {2n}\right) !} \] | Proof. This is a direct consequence of the theorem by choosing \( x = 0 \) for \( n \) even and \( x = 1/4 \) for \( n \) odd. Note that \( \mathop{\sum }\limits_{{k \geq 0}}{\left( -1\right) }^{k}/{\left( 2k + 1\right) }^{{2n} + 1} > 0 \) since it is an alternating series with decreasing terms. | Yes |
Corollary 9.1.22. As \( n \) tends to infinity, we have\n\n\[ \n{B}_{2n} \sim {\left( -1\right) }^{n - 1}\frac{2\left( {2n}\right) !}{{\left( 2\pi \right) }^{2n}}\;\text{ and } \n\]\n\n\[ \n{E}_{2n} \sim {\left( -1\right) }^{n}\frac{2\left( {2n}\right) !}{{\left( \pi /2\right) }^{{2n} + 1}}. \n\] | Proof. Clear since \( \mathop{\sum }\limits_{{k \geq 1}}1/{k}^{2n} \) and \( \mathop{\sum }\limits_{{k \geq 0}}{\left( -1\right) }^{k}/{\left( 2k + 1\right) }^{{2n} + 1} \) tend to 1 as \( n \rightarrow \infty \) . | No |
Corollary 9.1.23. If \( n \) is even we have\n\n\[ \mathop{\sup }\limits_{{x \in \mathbb{R}}}\left| {{B}_{n}\left( {\{ x\} }\right) }\right| = \left| {B}_{n}\right| \]\n\nand if \( n \) is odd we have\n\n\[ \mathop{\sup }\limits_{{x \in \mathbb{R}}}\left| {{B}_{n}\left( {\{ x\} }\right) }\right| \leq \frac{7\left| {B}_... | Proof. The first statement immediately follows from Theorem 9.1.20 and the fact that \( \left| {\cos \left( {2\pi kx}\right) }\right| \leq 1 \), with equality for all \( k \) if \( x = 0 \) . This proof is not valid for \( n \) odd. For \( n = 1 \) we have \( {B}_{1}\left( x\right) = x - 1/2 \), hence \( \mathop{\sup }... | Yes |
Theorem 9.2.2 (Euler-MacLaurin). Let \( a \) and \( b \) be two real numbers such that \( a \leq b \), and assume that \( f \in {C}^{k}\left( \left\lbrack {a, b}\right\rbrack \right) \) for some \( k \geq 1 \) . Then\n\n\[ \mathop{\sum }\limits_{\substack{{a < m \leq b} \\ {m \in \mathbb{Z}} }}f\left( m\right) = {\int ... | Proof. We give a clean proof, using (very little) the language of distributions, and explain very briefly afterward how to avoid it.\n\nBy the basic properties of Bernoulli polynomials, we know that \( {B}_{k}^{\prime }\left( {\{ t\} }\right) = \) \( k{B}_{k - 1}\left( {\{ t\} }\right) \) for \( k \geq 2 \), except for... | Yes |
Corollary 9.2.3. Let \( a \in \mathbb{R}, N \in {\mathbb{Z}}_{ \geq 0} \), and \( k \in {\mathbb{Z}}_{ \geq 1} \) . (1) If \( f \in {C}^{k}\left( \left\lbrack {a, N + a}\right\rbrack \right) \) we have \[ \mathop{\sum }\limits_{{m = 0}}^{{N - 1}}f\left( {m + a}\right) = {\int }_{a}^{N + a}f\left( t\right) {dt} + \matho... | Proof. For (1) we apply the theorem to \( a = 0 \) and \( b = N \), replace the function \( f\left( t\right) \) by \( f\left( {t + a}\right) \), and subtract \( f\left( {N + a}\right) - f\left( a\right) \) . | Yes |
Corollary 9.2.4. Let \( f \in {C}^{2k}\left( {\lbrack a,\infty \lbrack }\right) \) for some \( a \in \mathbb{R} \) . Assume that both the series \( \mathop{\sum }\limits_{{m \geq a}}f\left( m\right) \) and the integral \( {\int }_{a}^{\infty }f\left( t\right) {dt} \) converge, and that the derivatives \( {f}^{\left( 2j... | Proof. Immediate and left to the reader (Exercise 52). | No |
Corollary 9.2.6. Let \( k \geq 1 \), and let \( f \in {C}^{k}(\lbrack a,\infty \lbrack \) ).\n\n(1) Assume that the sign of \( {f}^{\left( k\right) }\left( t\right) \) is constant on \( \lbrack a,\infty \lbrack \) and that \( {f}^{\left( k - 1\right) }\left( t\right) \) tends to 0 as \( t \rightarrow \infty \) . There ... | Proof. By the Euler-MacLaurin formula, we have for all \( k \geq 1 \) \n\n\[ \mathop{\sum }\limits_{{m = 0}}^{{N - 1}}f\left( {m + a}\right) = {z}_{k}\left( {f, a, N}\right) + {\int }_{a}^{N + a}f\left( t\right) {dt} + \mathop{\sum }\limits_{{j = 1}}^{k}\frac{{B}_{j}}{j!}{f}^{\left( j - 1\right) }\left( {N + a}\right) ... | Yes |
Proposition 9.2.8. Keep the above assumptions on \( g \) and set \( f\left( x\right) = \) \( \mathcal{L}\left( g\right) \left( x\right) \) . For \( k \geq 1 \) we have\n\n(1)\n\n\[ \frac{{\left( -1\right) }^{k - 1}}{k!}{\int }_{a}^{N + a}{f}^{\left( k\right) }\left( t\right) {B}_{k}\left( {\{ t - a\} }\right) {dt} \]\n... | Proof. Immediate from the above remarks and left to the reader (Exercise 31). Note that the result is false for \( k = 0 \) . | No |
Corollary 9.2.9. Keep the above assumptions on \( g \) and assume that \( f \) and all its derivatives have constant sign and tend to 0 as \( t \rightarrow \infty \) . With the notation of Corollary 9.2.6 we have\n\n\[ z\left( {f, a}\right) = \frac{f\left( a\right) }{2} + {\int }_{a}^{\infty }{f}^{\prime }\left( t\righ... | Proof. Simply take \( k = 1 \) in the proposition. | No |
Proposition 9.2.10. (1) For \( \Re \left( s\right) > 0 \) and \( x > 0 \) we have\n\n\[ \zeta \left( {s, x + 1}\right) = \frac{{x}^{1 - s}}{s - 1} - s{\int }_{x}^{\infty }\frac{\{ t - x\} }{{t}^{s + 1}}{dt} = \frac{1}{\Gamma \left( s\right) }{\int }_{0}^{\infty }\frac{{t}^{s - 1}{e}^{-{xt}}}{{e}^{t} - 1}{dt}. \] | Proof. All the results except (5) and (6) are direct consequences of the definitions and of the proposition and its corollary. | No |
Proposition 9.2.11 (Abel-Plana). Assume that \( f \) is a holomorphic function on \( \Re \left( z\right) > 0 \), that \( f\left( z\right) = o\left( {\exp \left( {{2\pi }\left| {\Im \left( z\right) }\right| }\right) }\right) \) as \( \left| {\Im \left( z\right) }\right| \rightarrow \infty \) uniformly in vertical strips... | Proof. See Exercise 33. | No |
Proposition 9.2.12. For every \( k \geq 1 \) we have\n\n\[ \mathop{\sum }\limits_{{m = 1}}^{N}{m}^{k} = \frac{1}{k + 1}\left( {{N}^{k + 1} + \frac{k + 1}{2}{N}^{k} + \mathop{\sum }\limits_{{j = 2}}^{k}\left( \begin{matrix} k + 1 \\ j \end{matrix}\right) {B}_{j}{N}^{k + 1 - j}}\right) \]\n\n\[ = {N}^{k} + \frac{{B}_{k +... | Proof. Immediate application of Euler-MacLaurin with \( f\left( t\right) = {t}^{k} \) . The proposition is also easily proved directly using Proposition 9.1.3. | Yes |
Proposition 9.2.13. Let \( \alpha \in \mathbb{C} \) be different from -1 .\n\n(1) For every \( k > \Re \left( \alpha \right) + 1 \) such that \( k \geq 1 \) we have\n\n\[ \mathop{\sum }\limits_{{m = 1}}^{N}{m}^{\alpha } = \zeta \left( {-\alpha }\right) + \frac{{N}^{\alpha + 1}}{\alpha + 1} + \frac{{N}^{\alpha }}{2} + \... | Proof. The first statement follows directly from the Euler-MacLaurin formula and Proposition 9.2.5, apart from the determination of the constant. Fix some integer \( {k}_{0} > \Re \left( \alpha \right) + 1 \) such that \( {k}_{0} \geq 1 \), and let \( {f}_{\alpha }\left( t\right) = {t}^{\alpha } \). For all \( k \geq {... | Yes |
Proposition 9.2.14. (1) For \( k \geq 1 \) we have\n\n\[ \mathop{\sum }\limits_{{m = 1}}^{N}\frac{1}{m} = \log N + \gamma + \frac{1}{2N} - \mathop{\sum }\limits_{{j = 2}}^{k}\frac{{B}_{j}}{j{N}^{j}} + {R}_{k}\left( {-1, N}\right) ,\]\n\nwhere \( \gamma \) is Euler’s constant and\n\n\[ {R}_{k}\left( {-1, N}\right) = {\i... | Proof. (1) is again a direct application of Euler-MacLaurin and the definition of \( \gamma \), and (2) follows by choosing \( N = 1 \) . If we choose \( k = 1 \) in (2) of the preceding proposition with \( \alpha = - s \) we obtain\n\n\[ \zeta \left( s\right) = \frac{1}{s - 1} + \frac{1}{2} - s{\int }_{1}^{\infty }{t}... | Yes |
Proposition 9.3.1. Let \( \left\lbrack {a, b}\right\rbrack \) be a finite closed interval, and assume that \( f \in {C}^{k}\left( \left\lbrack {a, b}\right\rbrack \right) \) for some \( k \geq 1 \) . Then for any integer \( N \geq 1 \), if we set \( h = \) \( \left( {b - a}\right) /N \) we have\n\n\[{\int }_{a}^{b}f\le... | Proof. For \( t \in \left\lbrack {0, N}\right\rbrack \), set \( g\left( t\right) = f\left( {a + {ht}}\right) \) and apply the formula to the function \( g \) on the interval \( \left\lbrack {0, N}\right\rbrack \) . Since \( {g}^{\left( j\right) }\left( t\right) = {h}^{j}{f}^{\left( j\right) }\left( {a + {ht}}\right) \)... | Yes |
Lemma 9.4.3. We have\n\n\[ t{e}^{tx}\frac{\mathop{\sum }\limits_{{1 \leq r \leq m}}\chi \left( r\right) {e}^{-{rt}}}{1 - {e}^{-{mt}}} = \mathop{\sum }\limits_{{k \geq 0}}\frac{{B}_{k}\left( {{\chi }^{ - }, x}\right) }{k!}{t}^{k}. \] | Proof. Immediate and left to the reader. | No |
Proposition 9.4.4. We have \( {B}_{k}^{\prime }\left( {\chi, x}\right) = k{B}_{k - 1}\left( {\chi, x}\right) \) . | Proof. Clear since \( \left( {d/{dx}}\right) E\left( {\chi, t, x}\right) = {tE}\left( {\chi, t, x}\right) \) . | No |
Proposition 9.4.5. We have the following formulas:\n\n\[ \n{B}_{k}\left( {\chi, x}\right) = \mathop{\sum }\limits_{{j = 0}}^{k}\left( \begin{array}{l} k \\ j \end{array}\right) {B}_{j}\left( \chi \right) {x}^{k - j} = {m}^{k - 1}\mathop{\sum }\limits_{{0 \leq r < m}}\chi \left( r\right) {B}_{k}\left( \frac{x + r}{m}\ri... | Proof. The first formula follows from the identity \( E\left( {\chi, t, x}\right) = {e}^{tx}E\left( {\chi, t,0}\right) \) . The second follows from\n\n\[ \nE\left( {\chi, t, x}\right) = \frac{1}{m}\mathop{\sum }\limits_{{0 \leq r < m}}\chi \left( r\right) \frac{{mt}{e}^{\left( {\left( {x + r}\right) /m}\right) {mt}}}{{... | Yes |
Corollary 9.4.6. If \( x \in {\mathbb{Z}}_{ \geq 0} \) we have\n\n\[ \n{m}^{k - 1}\mathop{\sum }\limits_{{0 \leq r < m}}\chi \left( {x + r}\right) {B}_{k}\left( \frac{x + r}{m}\right) = {B}_{k}\left( \chi \right) + k\mathop{\sum }\limits_{{0 \leq r < x}}\chi \left( r\right) {r}^{k - 1}. \n\] | Proof. Indeed, for any function \( f \) and \( x \in \mathbb{Z} \geq 0 \) we have\n\n\[ \n\mathop{\sum }\limits_{{0 \leq r < m}}f\left( {x + r}\right) = \mathop{\sum }\limits_{{x \leq r < m + x}}f\left( r\right) = \mathop{\sum }\limits_{{0 \leq r < m}}f\left( r\right) + \mathop{\sum }\limits_{{0 \leq r < x}}\left( {f\l... | Yes |
Lemma 9.4.7. If \( m \mid M \) then\n\n\[ \n{B}_{k}\left( {\chi, x}\right) = {M}^{k - 1}\mathop{\sum }\limits_{{0 \leq r < M}}\chi \left( r\right) {B}_{k}\left( \frac{x + r}{M}\right) .\n\] | Proof. Write \( n = M/m \), and for \( 0 \leq r < M \) let \( r = {qm} + s \) with \( 0 \leq \) \( s < m \) and \( 0 \leq q < n \) . By the distribution formula for Bernoulli polynomials (Proposition 9.1.3) we have\n\n\[ \n{M}^{k - 1}\mathop{\sum }\limits_{{0 \leq r < M}}\chi \left( r\right) {B}_{k}\left( \frac{x + r}{... | Yes |
Proposition 9.4.8. We have\n\n\[ \n{B}_{k}\left( {\chi, x + m}\right) = {B}_{k}\left( {\chi, x}\right) + k\mathop{\sum }\limits_{{0 \leq r < m}}\chi \left( r\right) {\left( x + r\right) }^{k - 1}. \n\] | Proof. Follows from the formula\n\n\[ \nE\left( {\chi, t, x + m}\right) - E\left( {\chi, t, x}\right) = t\mathop{\sum }\limits_{{0 \leq r < m}}\chi \left( r\right) {e}^{\left( {x + r}\right) t}. \n\] | Yes |
Proposition 9.4.9. (1) We have\n\n\[ \n{B}_{k}\left( {\chi , - x}\right) = {\left( -1\right) }^{k}\left( {{B}_{k}\left( {{\chi }^{ - }, x}\right) + \chi \left( 0\right) k{x}^{k - 1}}\right) ,\n\]\n\nor equivalently,\n\n\[ \n{B}_{k}\left( {{\chi }^{ - }, x}\right) = {\left( -1\right) }^{k}{B}_{k}\left( {\chi , - x}\righ... | Proof. An easy computation shows that\n\n\[ \nE\left( {{\chi }^{ - }, t, x}\right) - E\left( {\chi , - t, - x}\right) = - \chi \left( 0\right) t{e}^{xt},\n\]\n\nwhich is clearly equivalent to the first formula, and the other statements follow by specializing to \( x = 0 \) . | Yes |
Proposition 9.4.11. The \( \chi \) -Bernoulli functions satisfy the following properties:\n\n(1) We have \( {B}_{0}\left( {\chi ,\{ x{\} }_{\chi }}\right) = {B}_{0}\left( \chi \right) = {S}_{0}\left( \chi \right) /m \) .\n\n(2) We have\n\n\[ \n{B}_{1}^{\prime }\left( {\chi ,\{ x{\} }_{\chi }}\right) = {B}_{0}\left( {\c... | Proof. All these properties are essentially clear from the definition and the basic properties of ordinary Bernoulli polynomials. For instance, let us prove (2) and (8). An easy exercise in distributions (Exercise 50) shows that\n\n\[ \n{\left\{ \frac{x + r}{m}\right\} }^{\prime } = \frac{1}{m} - \mathop{\sum }\limits_... | No |
Proposition 9.4.12. We have\n\n\[ \n{B}_{k}\left( {\chi ,\{ - x{\} }_{\chi }}\right) = {\left( -1\right) }^{k}{B}_{k}\left( {{\chi }^{ - },\{ x{\} }_{\chi }}\right) - \chi \left( x\right) {\delta }_{x,\mathbb{Z}}{\delta }_{k,1}, \]\n\nwhere \( {\delta }_{x,\mathbb{Z}} = 1 \) if \( x \in \mathbb{Z} \) and \( {\delta }_{... | Proof. Left to the reader (Exercise 51). Note that this generalizes Proposition 9.4.9. | No |
Proposition 9.4.13. For \( x \in {\mathbb{R}}_{ \geq 0} \) we have\n\n\[ \n{B}_{k}\left( {\chi ,\{ x{\} }_{\chi }}\right) = {B}_{k}\left( {\chi, x}\right) - k\mathop{\sum }\limits_{{1 \leq r \leq x}}{\chi }^{ - }\left( r\right) {\left( x - r\right) }^{k - 1}.\n\] | Proof. Denote by \( {R}_{k}\left( x\right) \) the right-hand side of this formula. We could show that \( {R}_{k}\left( x\right) \) satisfies the first five conditions of Proposition 9.4.11, but it is easier to reason directly. Set \( {C}_{k}\left( x\right) = \mathop{\sum }\limits_{{1 \leq r \leq x}}{\chi }^{ - }\left( ... | Yes |
Proposition 9.4.14. For \( n \geq 2 \) we have the Fourier expansion\n\n\[ \n{B}_{n}\left( {\chi ,\{ x{\} }_{\chi }}\right) = - \frac{n!{m}^{n - 1}}{{\left( 2i\pi \right) }^{n}}\mathop{\sum }\limits_{{k \in \mathbb{Z}, k \neq 0}}\frac{\tau \left( {\chi, k}\right) }{{k}^{n}}{e}^{{2i\pi kx}/m}.\n\]\n\nFor \( n = 1 \) thi... | Proof. We have seen above that the function \( {B}_{n}\left( {\chi ,\{ x{\} }_{\chi }}\right) \) is piecewise \( {C}^{\infty } \) and continuous for \( n \geq 2 \), with simple discontinuities at the integers if \( n = 1 \), and periodic of period (dividing) \( m \) . Thus for \( n \geq 2 \) we have \( {B}_{n}\left( {\... | Yes |
Proposition 9.4.15. As above, let \( \chi \) be a periodic arithmetic function of period (dividing) \( m \), let \( a \) and \( b \) be two real numbers such that \( a \leq b \), and assume that \( f \in {C}^{k}\left( \left\lbrack {a, b}\right\rbrack \right) \) for some \( k \geq 1 \) . Then\n\n\[ \mathop{\sum }\limits... | Proof. The Bernoulli functions have been defined exactly in order for this proposition to be valid, and as can clearly be seen, in the present context it would have been much better to choose \( {B}_{k}\left( {{\chi }^{ - }, x}\right) \) as definition of \( \chi \) -Bernoulli polynomials. By their basic properties, if ... | Yes |
Corollary 9.4.16. If \( N \in {\mathbb{Z}}_{ \geq 0} \) is such that \( m \mid N \) and if \( f \in {C}^{k}\left( \left\lbrack {0, N}\right\rbrack \right) \) we have for \( k \geq 1 \)\n\n\[ \n\mathop{\sum }\limits_{{0 \leq r < N}}\chi \left( r\right) f\left( r\right) = {B}_{0}\left( \chi \right) {\int }_{0}^{N}f\left(... | Proof. This follows from the above proposition and the formulas\n\n\[ \n{B}_{j}\left( {{\chi }^{ - },\{ 0{\} }_{\chi }}\right) = {B}_{j}\left( {\chi }^{ - }\right) = {\left( -1\right) }^{j}{B}_{j}\left( \chi \right) - \chi \left( 0\right) {\delta }_{j,1} \n\]\n\ncoming from Proposition 9.4.9. | Yes |
Corollary 9.4.17. If \( m \mid N \) and \( k \geq 0 \) we have\n\n\[ \mathop{\sum }\limits_{{0 \leq r < N}}\chi \left( r\right) {\left( x + r\right) }^{k} = \frac{{B}_{k + 1}\left( {\chi, N + x}\right) - {B}_{k + 1}\left( {\chi, x}\right) }{k + 1}. \] | Proof. Clear. | No |
Corollary 9.4.18. Assume in addition that \( f \in {C}^{k}(\lbrack 1,\infty \lbrack ) \), that both the series \( \mathop{\sum }\limits_{{r \geq 1}}\chi \left( r\right) f\left( r\right) \) and the integral \( {\int }_{1}^{\infty }f\left( t\right) {dt} \) converge, and that the \( {f}^{\left( j\right) }\left( N\right) \... | Proof. Immediate and left to the reader (Exercise 52). | No |
Lemma 9.5.2. (1) For any \( m \in {\mathbb{Z}}_{ \geq 0} \) we have\n\n\[ \mathop{\sum }\limits_{{0 \leq r < {mf}}}\chi \left( r\right) {r}^{k} \equiv m{S}_{k}\left( \chi \right) \left( {\;\operatorname{mod}\;f}\right) .\n\]\n\n(2) If, in addition, \( \gcd \left( {m, f}\right) = 1 \) and \( \chi \) is nontrivial, then\... | Proof. Immediate by writing \( r = {qf} + s \) or \( r = {qm} + s \) respectively, and left to the reader (Exercise 55). | No |
Lemma 9.5.4. Let \( \chi \) be any character modulo \( p \) for some odd prime \( p \), let \( o\left( \chi \right) \mid \left( {p - 1}\right) \) be the order of \( \chi \), and let \( K = \mathbb{Q}\left( \chi \right) = \mathbb{Q}\left( {\zeta }_{o\left( \chi \right) }\right) \) be the corresponding cyclotomic field.\... | Proof. Let \( L = \mathbb{Q}\left( {\zeta }_{p - 1}\right) \supset K \) . Since \( p \equiv 1\left( {{\;\operatorname{mod}\;p} - 1}\right) \), by Proposition 3.5.18 the prime \( p \) splits completely in \( L \) . Let \( \mathfrak{P} \) be some prime ideal of \( L \) above \( p \) . By definition of the Teichmüller cha... | Yes |
Theorem 9.5.5. Let \( p \) be an odd prime, let \( v \in {\mathbb{Z}}_{ \geq 1} \), let \( \chi \) be a primitive Dirichlet character of conductor \( f = {p}^{v} \), let \( o\left( \chi \right) \) be the order of \( \chi \), and let \( K = \mathbb{Q}\left( \chi \right) = \mathbb{Q}\left( {\zeta }_{o\left( \chi \right) ... | Proof. (1),(2), and (3). The case \( v = 1 \) is nothing else than Lemma 9.5.4, so assume \( v \geq 2 \) . Let \( r \) be coprime to \( p \) . Writing \( r \equiv {r}_{1}{r}_{2}\left( {\;\operatorname{mod}\;{p}^{v}}\right) \) with \( {r}_{1} = {r}^{{p}^{v - 1}} \), it is clear that \( {r}_{1} \) is unique modulo \( p \... | Yes |
Corollary 9.5.7. Let \( \chi \) be a nontrivial primitive Dirichlet character of conductor \( f \) and of order \( o\left( \chi \right) \). (1) If either \( f \) is not a prime power, or if \( f = {p}^{v} \) is an odd prime power and either \( o\left( \chi \right) \neq {p}^{v - 1}\left( {p - 1}\right) /\gcd \left( {p -... | Proof. Clear. | No |
Corollary 9.5.9. Let \( D \) be the discriminant of a quadratic field, and assume that either \( D \equiv 0\left( {\;\operatorname{mod}\;4}\right) \) or \( D > 0 \). (1) We have \[ \mathop{\sum }\limits_{{0 \leq r < \left| D\right| }}\left( \frac{D}{r}\right) {r}^{2} \equiv 0\left( {{\;\operatorname{mod}\;4}D}\right) ,... | Proofs. Immediate consequences of the theorem and of the fact that when \( \chi \) is an even character and \( k \geq 2 \) is even, then \( {S}_{k}\left( \chi \right) \equiv 0\left( {\;\operatorname{mod}\;8}\right) \), except when \( k = 2 \), in which case the congruence is only modulo 4 (Exercise 56). | No |
Corollary 9.5.10. Let \( \chi \) be a primitive character of conductor \( f \) such that \( 4 \mid f \), and let \( k \in {\mathbb{Z}}_{ \geq 0} \) . (1) If \( f \) is not a power of 2 we have \[ \mathop{\sum }\limits_{{0 \leq r < f/2}}\chi \left( r\right) {r}^{k} \equiv \left\{ \begin{array}{ll} 0\left( {\;\operatorna... | Proof. Assume first that \( \chi \left( {-1}\right) = {\left( -1\right) }^{k} \) . We have \[ {S}_{k}\left( \chi \right) = \mathop{\sum }\limits_{{0 \leq r < f/2}}\left( {\chi \left( r\right) {r}^{k} + \chi \left( {f - r}\right) {\left( f - r\right) }^{k}}\right) \] \[ \equiv 2\mathop{\sum }\limits_{{0 \leq r < f/2}}\c... | Yes |
Lemma 9.5.12. Let \( \\chi \) be a primitive Dirichlet character of conductor \( f \) and let \( p \) be a prime number. Assume that we are in case (2) of the preceding lemma, in other words that either \( f = 1 \), or \( f = {p}^{v} \) with \( p \) odd or with \( p = 2 \) and \( v = 2 \) . Then for all \( k \) such th... | Proof. Keep the notation of the preceding proof. Since \( p \\mid N \) it is immediate to check that \( {v}_{p}\\left( {{N}^{j - 1}/\\left( {j + 1}\\right) }\\right) \\geq 0 \) for \( j \\geq 2 \), and also for \( j = 1 \) if \( p \\neq 2 \) . Since from the preceding lemma we know that \( {v}_{p}\\left( {{B}_{j}\\left... | Yes |
Theorem 9.5.13. Let \( \chi \) be a primitive Dirichlet character of conductor \( f \) , denote as usual by \( o\left( \chi \right) \) the order of \( \chi \), and let \( K = \mathbb{Q}\left( \chi \right) = \mathbb{Q}\left( {\zeta }_{o\left( \chi \right) }\right) \) . For any \( k \geq 0 \) such that \( \chi \left( {-1... | Proof. All these results follow immediately from Lemma 9.5.12: for \( f = 1 \) we apply Lemma 9.5.4, for \( f = 4 \) we use the evaluation \( {S}_{k}\left( \chi \right) = 1 - {3}^{k} \), and for the other values of \( f \) we use Theorems 9.5.3,9.5.5, and 9.5.6. | No |
Corollary 9.5.15. For even \( k > 0 \) we have\n\n\[ \n{D}_{k} = \mathop{\prod }\limits_{{\left( {p - 1}\right) \mid k}}p \n\] | Proof. Indeed, from the theorem it is clear that \( {D}_{k} \) divides \( \mathop{\prod }\limits_{{\left( {p - 1}\right) \mid k}}p \) , but conversely for any prime \( p \) such that \( \left( {p - 1}\right) \mid k \) the theorem implies that \( {v}_{p}\left( {B}_{k}\right) = - 1 \), so that the product of such \( p \)... | Yes |
Corollary 9.5.16. (1) For even \( k > 0 \) we have \( 6 \mid {D}_{k} \) . | Proof. The fact that \( 6 \mid {D}_{k} \) is clear from the above corollary. | No |
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