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Theorem 16 (d'Alembert-Gauss). Every algebraic equation of degree \( \geq 1 \) has at least one complex root. | To see this, consider the function \( f\left( z\right) = 1/p\left( z\right) \) where \( p \) is a polynomial not vanishing anywhere. Since \( p\left( z\right) \) is analytic in \( \mathbb{C} \), so also is \( f \) (Chap. II, \( {\mathrm{n}}^{ \circ }{22} \), Theorem 17 - one may also invoke holomorphy, easier to prove ... | No |
This is the case of the Riemann function \( \zeta \left( s\right) = \sum 1/{n}^{s} \) in the open set \( \operatorname{Re}\left( s\right) > 1 \) where the series converges. | For every \( \sigma > 1 \), the series converges normally in the half plane \( \operatorname{Re}\left( s\right) \geq \sigma \) since there \( \left| {1/{n}^{s}}\right| \leq 1/{n}^{\sigma } \) . The function is thus holomorphic in \( \operatorname{Re}\left( s\right) > \sigma \) for every \( \sigma > 1 \), so in fact in ... | Yes |
Consider similarly the elliptic functions à la Weierstrass of Chap. II, \( {\mathrm{n}}^{ \circ }{23} \), the sums of the series \( \sum {\left( z - \omega \right) }^{-k} \) extended over the periods \( \left( {k > 3}\right) \), or, for \( k = 2 \), the modified series \( \wp \left( z\right) \) . | We have shown that these are analytic by proving by a routine calculation that, in every disc \( \left| z\right| < R \), they are the sum of a finite number of terms, corresponding to the periods situated in this disc, and of an explicitly calculated power series. The preceding theorem yields the result without any cal... | Yes |
Consider (Chap. IV, \( {\mathrm{n}}^{ \circ }{20} \) ) Euler’s formula\n\n\[ P\left( z\right) = \mathop{\prod }\limits_{{n \geq 1}}{\left( 1 - {z}^{n}\right) }^{-1} = \mathop{\sum }\limits_{{n \geq 0}}p\left( n\right) {z}^{n} \]\n\nwhich appears in the theory of partitions. | Theorem 18 shows that the product \( \prod \left( {1 - {z}^{n}}\right) \) is holomorphic in the disc \( D : \left| z\right| < 1 \) and never vanishes; the left hand side is therefore also holomorphic in \( D \), whence the existence of an expansion in power series in \( D \) . Since we have \( p\left( n\right) \geq 1 \... | No |
Let \( q \) be a constant complex such that \( \left| q\right| < 1 \) and let us consider the infinite product\n\n\[ f\left( z\right) = \prod \left( {1 + {q}^{n}z}\right) \]\n\nwhere the product is extended over all \( n \geq 1 \). | Since \( \sum \left| {q}^{n}\right| < + \infty \), Weierstrass’ theorem applies, the result being an entire function of \( z \) . It is clear that \( f\left( z\right) = \left( {1 + {qz}}\right) f\left( {qz}\right) \), so that the power series \( f\left( z\right) = \sum {a}_{n}{z}^{n} \) which represents \( f \) in the ... | Yes |
Consider the infinite product\n\n\[ f\left( z\right) = z\prod \left( {1 - {z}^{2}/{n}^{2}}\right) \]\n\nextended over \( n \geq 1 \) . It satisfies the hypotheses of the theorem, with \( G = \mathbb{C} \) , so represents an entire function having simple zeros at the \( n \in \mathbb{Z} \) and is nonzero elsewhere. | Theorem 18 shows that\n\n\[ {f}^{\prime }\left( z\right) /f\left( z\right) = 1/z + \sum {2z}/\left( {{z}^{2} - {n}^{2}}\right) = \pi \cdot \cot {\pi z} = {\left( \sin \pi z\right) }^{\prime }/\sin {\pi z}. \]\n\nIn the connected open set \( \mathbb{C} - \mathbb{Z} \) where it is defined, the holomorphic function \( f\l... | Yes |
Let us consider, with Weierstrass, the infinite product \( \prod \left( {1 - z/\omega }\right) \) extended over the periods \( \omega \in L \) . | \[ 1 - z/\omega = \exp \left( {-z/\omega - {z}^{2}/2{\omega }^{2} - \ldots }\right) = \] \[ = \;\exp \left( {-z/\omega - {z}^{2}/2{\omega }^{2}}\right) \exp \left( {-{z}^{3}/3{\omega }^{3} - \ldots }\right) \] with \( \left| {1 - \exp \left( {-{z}^{3}/3{\omega }^{3} - \ldots }\right) }\right| \leq M\left| {{z}^{3}/3{\o... | Yes |
Theorem 19. Let \( f \) be a regulated function on \( \mathbb{T} \). Then\n\n\[ f\left( u\right) = \mathop{\lim }\limits_{\substack{{z \rightarrow 1} \\ {\left| z\right| < 1} }}{H}_{f}\left( {zu}\right) = \mathop{\lim }\limits_{\substack{{r \rightarrow 1} \\ {r < 1} }}\mathop{\sum }\limits_{\mathbb{Z}}{r}^{\left| n\rig... | In this way we immediately recover Weierstrass' theorem: every continuous and periodic function \( f \) is the uniform limit of trigonometric polynomials of the same period. The preceding theorem, with \( J = \mathbb{T} \), shows in fact that \( {H}_{f}\left( {ru}\right) \) converges uniformly on \( \mathbb{T} \) to \(... | Yes |
Theorem 20. Every harmonic function \( H \) on a disc \( \left| z\right| < R \) has a series expansion of the form\n\n\[ H\left( z\right) = \sum {c}_{n}{r}^{\left| n\right| }{u}^{n} = {c}_{0} + \mathop{\sum }\limits_{{n > 0}}\left\lbrack {{c}_{n}{\left( x + iy\right) }^{n} + \overline{{c}_{n}}{\left( x - iy\right) }^{n... | with\n\n\[ {c}_{n}{r}^{\left| n\right| } = \int H\left( {ru}\right) {u}^{-n}{dm}\left( u\right) \]\n\nfor every \( r < R \) and every \( n \in \mathbb{Z} \).\n\nThere is no greater problem of convergence for the series (6) than for the power series of \( f \) : they converge normally in every disc of radius \( r < R \)... | Yes |
Theorem 21. Let \( G \) an open set in \( \mathbb{C} \) and \( \left( {H}_{n}\right) \) a sequence of harmonic functions in \( G \) which converges uniformly on every compact \( K \subset G \) to a limit function \( H \) . Then \( H \) is harmonic, and, for any \( p \) and \( q \), the partial derivatives \( {H}_{n}^{\... | We know thanks to Borel-Lebesgue (Chap. V, \( {\mathrm{n}}^{ \circ }6 \) ) that uniform convergence on every compact subset is a property of local character: to verify it for every compact \( K \subset G \) it is enough to show that, for every \( a \in G \), it holds on a closed disc of centre \( a \) .\n\nSo choose an... | Yes |
Theorem 22. Let \( f \) be a continuous function on \( \mathbb{T} \) . Then the function equal to\n\n(26.3)\n\n\[ \n{H}_{f}\left( z\right) = {\int }_{\mathbb{T}}\frac{1 - {\left| z\right| }^{2}}{{\left| z - u\right| }^{2}}f\left( u\right) {dm}\left( u\right) \n\]\n\nfor \( \left| z\right| < 1 \) and to \( f \) on \( \m... | Uniqueness follows from the maximum theorem; see the beginning of the preceding \( {\mathrm{n}}^{ \circ } \) . | No |
Theorem 24. Let \( f \) be a function defined and continuous on \( \mathbb{R} \) such that\n\n(i) the series \( \sum f\left( {x + n}\right) \) converges normally on every compact set,\n\n(ii) \( \sum \left| {\widehat{f}\left( n\right) }\right| < + \infty \) .\n\nThen \( f \) is absolutely integrable on \( \mathbb{R} \)... | In practice, the convergence of the series \( \sum f\left( {x + n}\right) \) is almost always obtained by estimating \( f\left( x\right) \) for \( \left| x\right| \) large. Assume for example\n\n(27.8)\n\n\[ f\left( x\right) = O\left( {\left| x\right| }^{-s}\right) \;\text{ at infinity, with }s > 1. \]\n\nThe continuou... | Yes |
Calculate the Fourier transform of the function \( f(t) = 1/(z + t)^s \) where \( z \) is a non real complex parameter and \( s \) an integer \( \geq 2 \), with \( \operatorname{Im}(z) > 0 \). | \[ \int \exp \left( -2\pi int \right) (z + t)^{-s} dt = \left\{ \begin{array}{ll} (-2\pi i)^s u^{s - 1} \exp (2\pi iuz) / (s - 1)! & \text{if } u > 0, \\ 0 & \text{if } u \leq 0 \end{array} \right. \] on condition that \( \operatorname{Im}(z) > 0 \). | Yes |
\[ \widehat{f}\left( y\right) = \int \exp \left( {-t\left| x\right| - {2\pi ixy}}\right) {dx} \] | \[ \widehat{f}\left( y\right) = {2t}/\left( {{t}^{2} + 4{\pi }^{2}{y}^{2}}\right) . \] | Yes |
Theorem 26. Let \( f \) be a continuous absolutely integrable function on \( \mathbb{R} \). Suppose that \( \widehat{f} \) is absolutely integrable. Then\n\n\[ f\left( x\right) = \int \widehat{f}\left( y\right) \mathbf{e}\left( {xy}\right) {dy}\;\text{ for every }x \in \mathbb{R}. \] | Note that the proof uses only the following facts: (i) formula (2), which we established using a \ | No |
Theorem 27. Let \( f \) be an absolutely integrable regulated function. Then\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}{\int }_{-N}^{N}\widehat{f}\left( y\right) \mathbf{e}\left( {xy}\right) {dy} = \frac{1}{2}\left\lbrack {f\left( {x + }\right) + f\left( {x - }\right) }\right\rbrack \]\n\n at every point whe... | And why has the unknown constant \( c \) transformed itself surreptitiously into \( \frac{1}{2} \) ? Because, if one applies the formula to a sufficiently accommodating function, one already knows (Theorem 25) that the right hand side of (15) is equal to \( f\left( x\right) \) . So the constant \( c \) has no choice \(... | No |
Let \( f \) be a regulated function such that \( \int \left| {{x}^{p}f\left( x\right) }\right| {dx} < + \infty \) . Then \( \widehat{f} \) is of class \( {C}^{p} \) and\n\n\[ \n{D}^{k}\widehat{f}\left( y\right) = \int {\left( -2\pi ix\right) }^{k}f\left( x\right) \mathbf{e}\left( {xy}\right) {dx}\;\text{ for every }k \... | Now suppose that \( f \) is of class \( {C}^{1} \) (or, more generally, a primitive of a regulated function) and that \( \int \left| {{f}^{\prime }\left( x\right) }\right| {dx} < \infty \) . Integrating by parts, one has, for \( y \neq 0 \) ,\n\n\[ \n{\int }_{-T}^{T}f\left( x\right) {e}^{-{2\pi ixy}}{dx} = {\left. f\le... | Yes |
Theorem 1. A holomorphic function \( f \) on a domain \( G \) has primitive on \( G \) if and only if its integral along any admissible path in \( G \) depends only on the latter's endpoints. | In particular, the integral of \( f \) along a closed path, i.e. such that \( \mu \left( 0\right) = \) \( \mu \left( 1\right) \), is zero. In fact this condition is sufficient for ensuring the existence of a primitive. Indeed, if\n\n\[ \n{\mu }_{1},{\mu }_{2} : \left\lbrack {0,1}\right\rbrack \rightarrow G \n\]\n\nare ... | Yes |
Theorem 3. Let \( G \) be a domain in \( \mathbb{C}, f \) a holomorphic function on \( G \) and \( {\mu }_{0},{\mu }_{1} \) two admissible paths in \( G \) . If one of the two following conditions is satisfied, then the integrals of \( f \) along \( {\mu }_{0} \) and \( {\mu }_{1} \) are equal:\n\n(a) there is a fixed-... | Exercise 2 (direct proof of Theorem 3). We keep the above construction and notation by, for example, supposing that \( \sigma \) is a fixed-endpoint homotopy; proving this result amounts to showing that integrals along \( {\gamma }_{p} \) and \( {\gamma }_{p + 1} \) are equal for all \( p \) . For this, take a closed p... | No |
Let \( f \) be a holomorphic function on a simply connected domain \( G \) ; suppose that \( f \) does not vanish on \( G \) . Then, there is a holomorphic function \( g \) on \( G \) such that \( {e}^{g\left( z\right) } = f\left( z\right) \) for all \( z \in G \) ; it is unique up to addition of a multiple of \( {2\pi... | Since \( f \) does not vanish on \( G \), the function \( {f}^{\prime }/f \) is defined and holomorphic on \( G \), and hence admits a primitive \( g \) ; then \( {\left( {e}^{g}\right) }^{\prime } = {g}^{\prime }{e}^{g} = {e}^{g}{f}^{\prime }/f \), i.e. \( {\left( {e}^{g}\right) }^{\prime }f - {e}^{g}{f}^{\prime } = 0... | Yes |
Theorem 5 (Cauchy’s residue formula). Let \( G \) be a domain in \( \mathbb{C}, S \) a closed discrete subset of \( G \) and \( f \) a holomorphic function on \( {G}^{\prime } = G - S \) . Then\n\n\[{\int }_{\mu }f\left( \zeta \right) {d\zeta } = {2\pi i}\mathop{\sum }\limits_{{a \in S}}{\operatorname{Ind}}_{\mu }\left... | The figure below is considered in the classical theory where \( S = \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) is finite. It contains the path \( \mu \) and a path \( \nu \) consisting, on the one hand, of\n\n\n\nFig... | Yes |
Theorem 8 (Dixon). Let \( G \) be a domain and \( \mu \) a closed path in \( G \) . The following statements are equivalent:\n\n(i) The integral along \( \mu \) of any holomorphic function on \( G \) is zero.\n\n(ii) The interior of \( \mu \) is contained in \( G \) . | (i) \( \Rightarrow \) (ii). For some \( a \notin G \), consider the function \( f\left( z\right) = 1/\left( {z - a}\right) \) ; by (i), its integral along \( \mu \) is zero, but, up to a factor \( {2\pi i} \), it is also the index of \( a \) with respect to \( \mu \) ; the latter is, therefore, zero, proving (ii).\n\n(... | No |
Theorem 9. Let \( I \) be an interval, \( \mu \) a measure on \( I, U \) an open subset of \( \mathbb{C} \) and \( f : I \times U \rightarrow \mathbb{C} \) a function satisfying the following conditions :\n\n(a) \( f \) is continuous on \( I \times U \) ,\n\n(b) \( f\left( {t, z}\right) \) is a holomorphic function of ... | It is sufficient to prove the statements with respect to \( r = 1 \) : the general case will follow by a repeated application of the result.\n\nFirst proof. Theorem 24 bis of Chap. V, \( §7 \) is analogous to the result we need to prove but based on different assumptions: \( {f}^{\prime } \) (instead of \( f \) ) was a... | Yes |
Theorem 10. On a simply connected domain, a real harmonic function is the real part of a holomorphic function and is unique up to the addition of a pure imaginary constant. | Uniqueness is obvious. Let \( u \) be a harmonic function; suppose there is a holomorphic function \( f = u + {iv} \) such that \( u = \operatorname{Re}f \) . Then\n\n\[ \n{f}^{\prime } = {D}_{1}u + i{D}_{1}v\;\text{ and }\;{D}_{1}v = - {D}_{2}u, \n\]\n\nand so \( {f}^{\prime } = {D}_{1}u - i{D}_{2}u \) . Conversely, s... | Yes |
Theorem 12 (Paley-Wiener). Let \( \varphi \) be an entire function. The following two conditions are equivalent :\n\n(i) There is a number \( a > 0 \) such that \( \varphi \) satisfies \( \left( {PW}\right) \) for all \( n \) ;\n\n(ii) \( \varphi \) is the complex Fourier transform of a \( {C}^{\infty } \) function van... | It suffices to prove that (i) \( \Rightarrow \) (ii). The function \( {z}^{n}f\left( z\right) \) being bounded on every horizontal and in particular on \( \mathbb{R} \), the Fourier transform\n\n\[ \widehat{f}\left( t\right) = \int f\left( x\right) \mathbf{e}\left( {-{tx}}\right) {dx} \]\n\nof \( f \) on \( \mathbb{R} ... | Yes |
Theorem 14. Let \( {\mathcal{S}}_{ + } = \mathcal{S}\left( {\mathbb{R}}_{ + }\right) \) be the set of functions defined and infinitely differentiable for \( x \geq 0 \) and that are together with their derivatives rapidly decreasing at infinity. For any \( f \in \mathcal{S}\left( {\mathbb{R}}_{ + }\right) \), the Melli... | (a) Assertions (i) and (ii) for \( \varphi = {\Gamma }_{f} \), where \( f \in \mathcal{S} \). Assertion (i) was proved before the theorem. So was the formula\n\n(13.13)\n\n\[ \n\operatorname{Res}\left( {\varphi , - k}\right) = {a}_{k} = {f}^{\left( k\right) }\left( 0\right) /k!.\n\]\n\n\( {}^{63} \) it is in fact bound... | Yes |
Lemma 1. Let \( f \) be a function defined and infinitely differentiable on an open interval \( 0 < x < b \) . \( f \) can be extended to an infinitely differentiable function on the interval \( 0 \leq x < b \) if and only if the following conditions hold:\n\n(a) \( f \) has an asymptotic expansion\n\n\[ f\left( x\righ... | First of all, the relation \( f\left( x\right) = {a}_{0} + {a}_{1}x + o\left( x\right) \) shows both that \( f\left( x\right) \) tends to \( {a}_{0} \) as \( x \) approaches 0 and that if we define \( f\left( 0\right) = {a}_{0} \), then the function \( f \) thus extended has a derivative equal to \( {a}_{1} \) at the o... | Yes |
Lemma 2. Let \( \varphi \) be a function defined and holomorphic on an open set\n\n\[ U : a < \operatorname{Re}\left( s\right) < b,\;\operatorname{Im}\left( s\right) > c \]\n\nand \( m \) be a real number. If the function \( {s}^{m}\varphi \left( s\right) \) is bounded at infinity on the closed vertical strip of finite... | To see this, argue as in \( {\mathrm{n}}^{ \circ }4 \) ,(iv). Take a closed strip\n\n\[ B : \operatorname{Im}\left( s\right) \geq {c}^{\prime } > c,{a}^{\prime } \leq \operatorname{Re}\left( s\right) \leq {b}^{\prime }\text{with}a < {a}^{\prime } < {b}^{\prime } < b \]\n\ncontained in \( U \) and choose some \( r > 0 \... | Yes |
Theorem 1. Let \( f \) be a \( {\mathbb{R}}^{q} \) -valued \( {C}^{s} \) map defined in the neighbourhood of \( 0 \) in \( {\mathbb{R}}^{p} \) and such that \( f\left( 0\right) = 0 \) . Suppose that the rank \( r \) of \( f \) is constant in the neighbourhood of 0 . Then there are \( {C}^{s} \) local charts \( \left( {... | To prove the theorem, first note that, up to a permutation of the canonical coordinates in \( {\mathbb{R}}^{p}\; \) and \( \;{\mathbb{R}}^{q},\;D\left( {{f}^{1},\ldots ,{f}^{r}}\right) /D\left( {{x}^{1},}\right. \) \( \left. {\ldots ,{x}^{r}}\right) \neq 0 \) may be assumed to hold at 0, hence also in a neighbourhood o... | Yes |
Theorem 3. Let \( G \) be a domain in a Cartesian space \( E \) and \( \omega \) a closed differential form of class \( {C}^{1} \) on \( G \) . The integrals of \( \omega \) along the two admissible paths \( {\mu }_{0} \) and \( {\mu }_{1} \) in \( G \) are equal if one of the following conditions holds:\n\n(a) There i... | From this, we can deduce that if all closed paths in \( G \) are homotopic to a constant path, then any closed form on \( G \) has a primitive; a domain with this property is said to be simply connected.\n\nHomotopy being an equivalence relation, it is then obvious that condition (b) of the theorem always holds. The sa... | Yes |
Theorem 4. Let \( U \subset {\mathbb{R}}^{n} \) be an open bounded set, \( A \) its closure and \( \varphi \) a diffeomorphism of class \( {C}^{1} \) from \( A \) to a compact subset \( B \subset {\mathbb{R}}^{n} \) . Suppose that the borders of \( A \) and \( B \) have measure zero. Then, for any integrable \( {}^{39}... | We will prove the formula for continuous functions. By \( {\mathrm{n}}^{ \circ }{10} \) and 11 of Chapter V, \( §2 \) it can then be immediately generalized to lsc (resp. usc) functions; for this, simply observe that if there is an increasing (resp. decreasing) philtre \( \Phi \) for continuous functions, then the func... | No |
Lemma 1. Let \( U \) be an open subset of \( {\mathbb{R}}^{n} \) containing 0 and \( \psi : U \rightarrow {\mathbb{R}}^{n} \) a \( {C}^{1} \) map such that \( \psi \left( 0\right) = 0,{\psi }^{\prime }\left( 0\right) = 1 \) . Given a number \( q \) such that \( 0 < q < 1 \) , let \( r \) be a number \( > 0 \) such that... | The derivative \( {\psi }^{\prime }\left( x\right) \) being a continuous function of \( x \) equal to 1 at \( x = 0 \) , the existence of \( r \) for given \( q > 0 \) is obvious. Having said that, suppose \( x \in K\left( r\right) \), so that \( {tx} \in K\left( r\right) \) for \( 0 \leq t \leq 1 \) . The derivative o... | Yes |
For all \( q > 0 \), there exists \( r > 0 \) such that, for \( x, y \in A \), \[ \left| {x - y}\right| \leq r \Rightarrow \begin{Vmatrix}{{\varphi }^{\prime }\left( x\right) - {\varphi }^{\prime }\left( y\right) }\end{Vmatrix} \leq q/\begin{Vmatrix}{{\varphi }^{\prime }{\left( x\right) }^{-1}}\end{Vmatrix}. \] | The map \( x \mapsto {\varphi }^{\prime }\left( x\right) \) being continuous on \( A \) and \( {\varphi }^{\prime }\left( x\right) \) being invertible for all \( x \in A \), the map \( x \mapsto {\varphi }^{\prime }{\left( x\right) }^{-1} \) is also continuous on \( A \) : indeed Cramer’s formulas \( {}^{41} \) tell us... | Yes |
Lemma 1. Let \( A \) and \( B \) be two disjoint closed subsets in a metric space \( X \) . There is a function \( f \) defined and continuous on \( X \) satisfying \( f\left( x\right) = 1 \) on \( A, f\left( x\right) = 2 \) on \( B \) and \( 1 \leq f\left( x\right) \leq 2 \) everywhere. | The lemma is trivial if \( A \) is empty (take \( f = 2 \) everywhere) or if \( B \) is empty (take \( f = 1 \) everywhere). Otherwise, choose a distance function \( d\left( {x, y}\right) \) defining the topology on \( X \) and set\n\n\[ f\left( x\right) = \inf \left\lbrack {d\left( {x, A}\right) ,{2d}\left( {x, B}\rig... | Yes |
Lemma 2. Let \( U \) be an open set and \( A \) a closed one contained in \( U \) . There is an open set \( V \) such that \( A \subset V \subset \bar{V} \subset U \) . If \( X \) is locally compact \( {}^{69} \) and \( A \) is compact, \( \bar{V} \) may be assumed to be compact. | Here too, the first statement is trivial if \( U = X \) (take \( V = X \) ) or if \( A \) is empty (take \( V = \{ x\} \) with \( x \in U \) ). Otherwise, set \( B = X - U \), choose a function \( f \) by lemma 1 and take \( V = \{ f\left( x\right) < 3/2\} \), an open set containing \( A \) trivially and whose closure,... | No |
Lemma 3. Let \( {U}_{0},\ldots ,{U}_{n} \) be open sets and \( X \) their union. There exist open sets \( {V}_{0},\ldots ,{V}_{n} \) whose union is \( X \) and such that \( {\bar{V}}_{i} \subset {U}_{i} \) for all \( i \) . | The \( {V}_{p} \) are constructed by induction on \( p \) by requiring them to also satisfy\n\n\[ \n{V}_{0} \cup \ldots \cup {V}_{p} \cup {U}_{p + 1} \cup \ldots \cup {U}_{n} = X.\n\]\n\nAs \( {V}_{0} \) only needs to satisfy\n\n\[ \nX - \left( {{U}_{1} \cup \ldots \cup {U}_{n}}\right) \subset {V}_{0} \subset {\bar{V}}... | No |
Lemma 4. Let \( X \) be a locally compact metric space, \( A \) a compact subset of \( X \) and \( {\left( {U}_{i}\right) }_{1 \leq i \leq n} \) a finite open covering of \( A \) . Then, there exist positive valued continuous functions \( {f}_{i} \) on \( X \), with compact support and such that\n\n\[ \operatorname{Sup... | Set \( {U}_{0} = X - A \) . Lemma 3 gives open sets \( {V}_{i}\left( {0 \leq i \leq n}\right) \) covering \( X \) and such that \( {V}_{i} \subset \overline{{V}_{i}} \subset {U}_{i} \) . For \( 0 \leq i \leq n \), lemma 1 proves the existence of continuous functions \( {g}_{i} \) on \( X \), with values in \( \left\lbr... | Yes |
Theorem 1. Let \( X \) be a compact Riemann surface.\n\n(a) Any function defined and holomorphic on \( X \) is a constant. | (a) is obvious: a function \( h \) everywhere holomorphic reaches its maximum somewhere, and so is constant in the neighbourhood of its maximum. As \( X \) is by definition connected, classical arguments apply verbatim. (Corollary: the only entire elliptic functions on \( \mathbb{C} \) are the constants). | No |
Lemma 2. Let \( P\left( {X, Y}\right) \) be a polynomial with complex coefficients and \( E \subset \) \( {\mathbb{C}}^{2} \) the set of simple points \( {}^{7} \) of the curve \( P\left( {z,\zeta }\right) = 0 \) . Then \( E \) is a submanifold of \( {\mathbb{C}}^{2} = {\mathbb{R}}^{4} \) and, for all \( \left( {a,\alp... | Equivalently, there exists a unique holomorphic function \( f \) on \( V \) satisfying\n\n(2.3)\n\n\[ f\left( a\right) = \alpha \& P\left\lbrack {z, f\left( z\right) }\right\rbrack = 0\text{ for all }z \in V. \]\n\n\( f \) is called a local uniform branch at \( a \) of the algebraic function defined by \( P \) .\n\nSin... | Yes |
Choose \( B = {\mathbb{C}}^{ * }, X = \mathbb{C} \) and \( p \) to be the map\n\n\[ \mathbf{e} : \zeta \mapsto \exp \left( {2\pi i\zeta }\right) \]\n\nfrom \( X \) onto \( B \) . As \( {\mathbf{e}}^{\prime }\left( \zeta \right) \neq 0 \) everywhere, \( p \) is a local homeomorphism (Chap. VIII, \( {\mathrm{n}}^{ \circ ... | The periodicity of the exponential function shows that\n\n\[ {p}^{-1}\left( D\right) = \mathop{\bigcup }\limits_{\mathbb{Z}}{D}^{\prime } + n = \bigcup {D}_{n}, \]\n\nand if \( {D}^{\prime } \) (i.e. \( D \) ) is sufficiently small. the translates \( {D}_{n} \) of \( {D}^{\prime } \) are pairwise disjoint and homeomorp... | Yes |
Theorem 2. Let \( \left( {X, B, p}\right) \) be a covering and \( \gamma : I \rightarrow B \) a path in \( B \) . There is a unique lifting \( \mu \) from \( \gamma \) to \( X \) with a given initial point. If two paths \( {\gamma }_{0} \) and \( {\gamma }_{1} \) in \( B \) are fixed-endpoint homotopic and if \( {\mu }... | We first prove the uniqueness of \( \mu \) . In the neighbourhood of some \( {t}_{0} \in I \) , a lifting \( \mu \left( t\right) \) of \( \gamma \), being a continuous function of \( t \), takes its values in a neighbourhood of \( \mu \left( {t}_{0}\right) \) mapped homeomorphically by \( p : X \rightarrow B \) onto a ... | No |
Corollary 1. Let \( \left( {X, B, p}\right) \) be a covering and suppose that \( X \) is connected and simply connected. \( {}^{11} \) Let \( {\gamma }_{0} \) and \( {\gamma }_{1} \) be two paths in \( B \) with the same endpoints and let \( {\mu }_{0} \) and \( {\mu }_{1} \) be liftings to \( X \) of \( {\gamma }_{0} ... | If the condition holds, the two liftings are homotopic, since \( X \) is simply connected, and thus that is also the case of the given paths in \( B \) . The converse, which makes no assumptions on \( X \), is the second statement of theorem 2 . | No |
For any closed path \( \gamma \) in \( {\mathbb{C}}^{ * } \), there is a unique integer \( n \) such that \( \gamma \) is homotopic to \( n\mathbf{u} : t \mapsto \mathbf{e}\left( {nt}\right) \) . | Indeed, the map \( \mathbf{e} : \mathbb{C} \rightarrow {\mathbb{C}}^{ * } \) transforms \( \mathbb{C} \) into a connected and simply connected covering space of \( {\mathbb{C}}^{ * }\left\lbrack {\left( \mathrm{i}\right) \text{, Example 1}}\right\rbrack \) . A lifting of \( \gamma \) is a path \( \mu \) in \( \mathbb{C... | Yes |
Theorem 3. Every covering \( \left( {X, B, p}\right) \) of a simply connected and locally connected space is trivial. | It all amounts to showing the existence of a global section with given value \( \alpha \in X \) at a given point \( a \in B \) . To define it at an arbitrary \( z \in B \), connect \( a \) to \( z \) by a path \( \gamma \) and consider the lifting \( \mu \) of \( \gamma \) with initial point \( \alpha \) . If \( \gamma... | Yes |
Theorem 4. Every connected and locally simply connected space \( B \) has a connected and locally simply connected covering \( \left( {X, B, p}\right) \) . It is unique up to isomorphism. If \( \left( {Y, B, q}\right) \) is a connected covering of \( X \) and if \( \alpha \in X \) and \( \beta \in Y \) are such that \(... | (a) Choose a point \( a \in B \) and consider the set of all paths \( \gamma : I \rightarrow B \) with initial point \( a \) . Two homotopic (understood to be fixed-endpoint homotopic) paths are considered to be equivalent. Let \( X \) be the set of classes of these paths. Define \( p : X \rightarrow B \) by associatin... | No |
Theorem 5. Let \( \left( {Y,{D}^{ * }, q}\right) \) be a connected covering of order \( k < + \infty \) of the pointed disc \( {D}^{ * } : 0 < \left| z\right| < 1 \) . Then there is a holomorphic function \( \varphi \) in \( Y \) such that (i) \( \zeta \mapsto \varphi \left( \zeta \right) \) is a conformal representati... | This means that \( \left( {Y,{D}^{ * }, q}\right) \) is isomorphic to the covering of \( {D}^{ * } \) obtained by taking \( Y = {D}^{ * } \) and \( q\left( z\right) = {z}^{k} \) [(i), Example 2], or to the covering obtained by constructing the Riemann surface of the algebraic function \( {z}^{1/k} \) using the method o... | No |
Let \( {\zeta }_{i}\left( {1 \leq i \leq n}\right) \) be the roots of an equation\n\n\[{\zeta }^{n} + {c}_{1}{\zeta }^{n - 1} + \ldots + {c}_{n} = 0\]\n\nwith complex coefficients. Then\n\n\[\sup \left| {\zeta }_{i}\right| \leq \max \left( {1,\left| {c}_{1}\right| + \ldots + \left| {c}_{n}\right| }\right) . | Let \( M \) be the left hand side of (6). Each root of (5) satisfies\n\n\[{\left| {\zeta }_{i}\right| }^{n} \leq \left| {c}_{1}\right| \cdot {\left| {\zeta }_{i}\right| }^{n - 1} + \ldots + \left| {c}_{n}\right| \leq \left| {c}_{1}\right| \cdot {M}^{n - 1} + \ldots + \left| {c}_{n}\right| ,\]\n\nwhence\n\n\[{M}^{n} \le... | Yes |
Theorem 7. The Riemann surface of an irreducible algebraic equation is connected. | It suffices to prove this for the open subset \( X \) of \( \widehat{X} \) because the point \( {\eta }_{Y} \) adjoined to \( X \) can obviously be connected to points of \( X \) by paths.\n\nAs seen in section (ii) of the previous \( {\mathrm{n}}^{ \circ } \), like \( X \), any connected component \( {X}^{\prime } \) ... | No |
Theorem 8. All meromorphic functions on \( \widehat{X} \) are rational functions of \( z \) and \( \zeta \) . | We will assume a result from the general theory of commutative field extensions though it is not hard to show. Otherwise the proof given will be complete.\n\nLet \( \varphi \) be a meromorphic function on \( \widehat{X} \) . It has finitely many poles projecting onto points \( {a}_{i} \in \widehat{\mathbb{C}} \) . Let ... | No |
Lemma 2. Let \( M \) be a commutative field, \( K \) a subfield of \( M \) and \( \zeta \) an element of \( M \) satisfying an irreducible algebraic equation of degree \( n \) over \( K \). Then the subfield \( L \) of \( M \) generated by \( K \) and \( \zeta \) has dimension \( n \) over \( K \) and admits \( 1,\zeta... | Since \( {\zeta }^{n} \) is a linear combination of \( 1,\zeta ,\ldots ,{\zeta }^{n - 1} \) with coefficients in \( K \), the same is true for \( {\zeta }^{p} \) for all \( p > n \) :\n\n\[ \n{\zeta }^{n + 1} = \zeta \left( {{c}_{0} + \ldots + {c}_{n - 1}{\zeta }^{n - 1}}\right) = {c}_{0}\zeta + \ldots + {c}_{n - 1}{\z... | Yes |
Lemma 3. Let \( M \) be a commutative field of characteristic \( 0, K \) a subfield of \( M \) and \( n \) an integer \( \geq 1 \) . Suppose that all \( x \in M \) satisfy an algebraic equation of degree \( \leq n \) with coefficients in \( K \) . Then, the dimension of \( M \) as a vector space over \( K \) is \( \leq... | This lemma is itself based on the primitive element theorem due to Dedekind for the field of algebraic numbers and valid for all fields of characteristic 0: if all \( x \in M \) are algebraic over \( K \), then for any finitely number of elements \( {x}_{1},\ldots ,{x}_{p} \in M \), there exists \( x \) such that the s... | No |
Lemma 1. Let \( \mu \) be a positive linear functional on \( L\left( X\right) \) . Then\n\n(1.1)\n\n\[ \left| {\mu \left( f\right) }\right| \leq \mu \left( \left| f\right| \right) \text{ for all }f \in L\left( X\right) . \] | This is obvious if \( f \) is real since \( f = {f}^{ + } - {f}^{ - },\left| f\right| = {f}^{ + } + {f}^{ - } \) . If \( f = g + {ih} \) is complex-valued, multiplying \( f \) by a complex factor with absolute value 1, \( \mu \left( f\right) \) may be assumed to be real and \( > 0 \) . Then \( \mu \left( h\right) = 0 \... | Yes |
For any compact set \( K \subset X \), there is a constant \( {M}_{K}\left( \mu \right) \) such that\n\n\[ \left| {\mu \left( f\right) }\right| \leq {M}_{K}\left( \mu \right) \parallel f\parallel \;\text{ for all }f \in L\left( {X, K}\right) . \] | Indeed, there is a positive-valued continuous function \( p \) on \( X \) equal to 1 on \( K \) and zero outside a compact set \( {K}^{\prime } \supset K \) . Then \( \left| {f\left( x\right) }\right| \leq \parallel f\parallel p\left( x\right) \) for all \( f \in L\left( {X, K}\right) \), and so\n\n\[ \left| {\mu \left... | Yes |
Lemma 3 (Dini’s Theorem). Let \( \Phi \subset {L}_{\mathbb{R}}\left( X\right) \) be an increasing philtre \( \varphi = \sup \left( \Phi \right) \) . Suppose that \( \varphi \in L\left( X\right) \) . Then\n\n\[ \mathop{\lim }\limits_{\Phi }\parallel \varphi - f{\parallel }_{X} = 0\;\text{ and }\;\mu \left( \varphi \righ... | The proof is the same as in Chap. V, \( {\mathrm{n}}^{ \circ }{10} \) . We first suppose that \( \Phi \subset \) \( {L}_{ + }\left( X\right) \) . Let \( r \) be a number \( > 0 \) and \( K \) the support of \( \varphi \) . As \( \varphi \left( x\right) = \sup f\left( x\right) \) is everywhere \( < + \infty \), for all ... | Yes |
Lemma 4. Let \( \Phi \) be an increasing philtre of continuous functions with compact support and \( \varphi \) its upper envelope. Then\n\n(1.6)\n\n\[ \n{\mu }^{ * }\left( \varphi \right) = \mathop{\sup }\limits_{\Phi }\mu \left( f\right) = \mathop{\lim }\limits_{\Phi }\mu \left( f\right) .\n\] | The first expression is obviously greater than the second one. So, it suffices to prove the reverse inequality. For any \( g \in {L}_{\inf }\left( X\right) \), let \( {\Phi }_{g} \subset {L}_{\mathbb{R}}\left( X\right) \) be the set of functions of the form \( \inf \left( {f, g}\right) \) where \( f \in \Phi \) . Since... | Yes |
Theorem 1. The function \( \varphi \mapsto {\mu }^{ * }\left( \varphi \right) ,\varphi \in \mathcal{I} \), has the following properties:\n\n(i) Additivity:\n\n\[ \n{\mu }^{ * }\left( {\varphi + \psi }\right) = {\mu }^{ * }\left( \varphi \right) + {\mu }^{ * }\left( \psi \right) \n\] \n\n(ii) Passage to the limit for in... | The proofs are the same as in Chap. V, \( {\mathrm{n}}^{ \circ }{10} \) and 11. To prove (i), choose two increasing philtres \( \Phi \) and \( \Psi \) in \( {L}_{ + }\left( X\right) \) converging to \( \varphi \) and \( \psi \) and note that the functions \( f + g \), with \( f \in \Phi \) and \( g \in \Psi \), form an... | Yes |
Theorem 3. All subsets of a null set are null; the finite or countable union of a family of null sets is null. | Obvious. | No |
Theorem 4. Let \( f \) be a function with values in \( \left\lbrack {0, + \infty }\right\rbrack \) ; then\n\n(2.8)\n\n\[{\mu }^{ * }\left( f\right) = 0 \Leftrightarrow f\left( x\right) = 0\text{ ae. }\]\n\n(2.9)\n\n\[{\mu }^{ * }\left( f\right) < + \infty \Rightarrow f\left( x\right) < + \infty \text{ ae. }\] | Supposing that \( f \geq 0 \) and \( {\mu }^{ * }\left( f\right) = 0 \), let us consider the set \( {N}_{p} \) of \( x \) where \( f\left( x\right) > 1/p \) and its characteristic function \( {\chi }_{p} \) . Then \( f \geq {\chi }_{p}/p \), whence\n\n\[{\mu }^{ * }\left( {N}_{p}\right) = {\mu }^{ * }\left( {\chi }_{p}... | Yes |
Theorem 5. Let \( \sum {f}_{n}\left( x\right) \) be a series of complex-valued functions such that \( \sum {N}_{p}\left( {f}_{n}\right) < + \infty \) for given \( p \geq 1 \) . Then\n\n(3.8)\n\n\[ \sum \left| {{f}_{n}\left( x\right) }\right| < + \infty \text{ ae. } \]\n\nand any function \( f \) such that\n\n(3.9)\n\n\... | Let us start with the case of a series \( \sum {f}_{n} \) whose terms are \( \geq 0 \) and denote the partial sums by \( {s}_{n} \) . The increasing sequence of functions \( {s}_{n}^{p} \) converges to \( {f}^{p} \), where for all \( x \), we set \( f\left( x\right) = \sum {f}_{n}\left( x\right) \leq + \infty \) . Then... | Yes |
Lemma 1. If \( f \) and \( g \) are integrable, so is \( {\alpha f} + {\beta g} \) for all \( \alpha ,\beta \in \mathbb{C} \), and\n\n\[ \mu \left( {{\alpha f} + {\beta g}}\right) = {\alpha \mu }\left( f\right) + {\beta \mu }\left( g\right) . | This follows from the inequality\n\n(4.4)\n\n\[ {N}_{1}\left\lbrack {\left( {f + g}\right) - \left( {{f}_{n} + {g}_{n}}\right) }\right\rbrack \leq {N}_{1}\left( {f - {f}_{n}}\right) + {N}_{1}\left( {f - {f}_{n}}\right) \]\n\nand from the linearity of the integral of continuous functions. | No |
Lemma 2. Let \( \left( {f}_{n}\right) \) be a sequence of functions in \( {L}^{p} \) and \( f \) a function such that \( \lim {N}_{p}\left( {f - {f}_{n}}\right) = 0 \) . Then \( f \) is in \( {L}^{p} \) . If \( p = 1 \), then | (4.6)\n\n\[\n\mu \left( f\right) = \lim \mu \left( {f}_{n}\right)\n\]\n\nObvious. | No |
Theorem 6. Let \( \left( {f}_{n}\right) \) be a sequence of functions in \( {L}^{p} \) such that \( \sum {N}_{p}\left( {f}_{n}\right) < \) \( + \infty \) . Then \( \sum \left| {{f}_{n}\left( x\right) }\right| < + \infty \) ae. Any function \( f \) satisfying \( f\left( x\right) = \overline{\sum }{f}_{n}\left( x\right) ... | This is theorem 5 applied to functions in \( {L}^{p} \) . By lemma 2, the (class of the) limit function \( f \) is still in \( {L}^{p} \) and (8) follows from (4) and (7). | Yes |
Theorem 7 (Riesz-Fischer). The normed vector space \( {L}^{p} \) is complete. \( A \) sequence of functions \( {f}_{n}\left( x\right) \) in \( {L}^{p} \) converging in mean to a limit \( f \) contains a subsequence converging almost everywhere to \( f \) . If, moreover, the sequence \( {f}_{n}\left( x\right) \) converg... | Being a closed subspace of a complete space, \( {L}^{p} \) is complete. The second proposition of the statement is here too theorem 5 in the special case of \( {L}^{p} \) . | No |
For any \( f \in {L}^{p} \), there is a series of continuous functions with compact support such that\n\n\[ \sum {N}_{p}\left( {f}_{n}\right) < + \infty ,\;f\left( x\right) = \sum {f}_{n}\left( x\right) \text{ ae. }, \]\n\n\[ \lim {N}_{p}\left( {f - {f}_{1} - \ldots - {f}_{n}}\right) = 0. \] | If \( f \) is real-valued, the \( {f}_{n} \) may be assumed to be real. Writing \( {\varphi }^{\prime }\left( x\right) = \) \( \sum {f}_{n}^{ + }\left( x\right) ,{\varphi }^{\prime \prime }\left( x\right) = \sum {f}_{n}^{ - }\left( x\right) \) defined lsc functions with values in \( \left\lbrack {0, + \infty }\right\rb... | Yes |
For all \( f \in {L}^{p} \), there is a sequence of functions \( {f}_{n} \in L\left( X\right) \) and a function \( F \in {L}^{p} \) such that\n\n\[ f\left( x\right) = \lim {f}_{n}\left( x\right) \text{ ae.,}\;\left| {{f}_{n}\left( x\right) }\right| \leq \left| {F\left( x\right) }\right| \text{ ae. } \] | To prove the first point, it suffices to consider the partial sums \( {s}_{n}\left( x\right) \) of the series whose existence is ensured by corollary 1 and to choose \( F\left( x\right) = \) \( \sum \left| {{f}_{n}\left( x\right) }\right| \) . By theorem 6 applied to the \( \left| {f}_{n}\right|, F \in {L}^{p} \) . | No |
Lemma 3. A lsc or usc function \( \varphi \) is integrable if and only if \( {N}_{1}\left( \varphi \right) < + \infty \) . | As \( - \varphi \) is lsc if \( \varphi \) is usc, we need only consider the case when \( \varphi \) is lsc and show that condition \( {N}_{1}\left( \varphi \right) < + \infty \) is sufficient. As \( \varphi \) is lsc, \( {\varphi }^{ + } \) is lsc, \( {\varphi }^{ - } \) is usc and these two functions satisfy the cond... | Yes |
Lemma 4. A function \( f \) with values in \( \left\lbrack {-\infty , + \infty }\right\rbrack \) is integrable if and only if, for all \( r > 0 \), there exist integrable functions \( \varphi \) and \( \psi \), lsc and usc respectively, such that\n\n\[ \psi \leq f \leq \varphi \;\& \;{\mu }^{ * }\left( {\varphi - \psi ... | If \( f \) is integrable, there is a function \( u \in {L}_{\mathbb{R}}\left( X\right) \) such that \( {\mu }^{ * }\left( \left| {f - u}\right| \right) < r \) , hence a lsc function \( \theta \) such that\n\n\[ \left| {f - u}\right| \leq \theta ,\;{\mu }^{ * }\left( \theta \right) \leq r. \]\n\nAs \( - \theta \leq f - ... | Yes |
Lemma 1. The upper (resp. lower) envelope of a finite family of functions of \( {L}^{p} \) is in \( {L}^{p} \) . | For real functions, inequality \( \left| {{f}^{ + } - {g}^{ + }}\right| \leq \left| {f - g}\right| \) shows that \( {N}_{p}\left( {{f}^{ + } - {g}^{ + }}\right) \leq \) \( {N}_{p}\left( {f - g}\right) \) . If \( f \) is in \( {\mathcal{L}}^{p} \), so are \( {f}^{ + } \) and \( {f}^{ - } \) . We conclude the proof using... | No |
Theorem 8. Let \( \left( {f}_{n}\right) \) be an increasing sequence of positive functions in \( {L}^{p}.f = \sup {f}_{n} = \lim {f}_{n} \) is in \( {L}^{p} \) if and only if \( \sup {N}_{p}\left( {f}_{n}\right) < + \infty \) . Then \( \lim {N}_{p}\left( {f - {f}_{n}}\right) = 0. \) | The condition is clearly necessary since \( 0 \leq {f}_{n} \leq f \) for all \( n \) . To obtain the converse, it suffices to show that the sequence \( \left( {f}_{n}\right) \) satisfies Cauchy’s criterion in \( {L}^{p} \) . This is easy if \( p = 1 \) . Indeed, since \( {f}_{j} - {f}_{i} \) is integrable and positive ... | Yes |
Lemma 2. Any decreasing sequence \( \left( {f}_{n}\right) \) of positive functions in \( {L}^{p} \) converge in mean. | Apply theorem 8 to functions \( {f}_{1} - {f}_{n} \) . | No |
Lemma 3. The lower envelope of a countable family \( \left( {f}_{n}\right) \) of positive functions in \( {L}^{p} \) is in \( {L}^{p} \) . | Apply the previous result to the functions \( \inf \left( {{f}_{1},\ldots ,{f}_{n}}\right) \) ; we already know (lemma 1) they are in \( {L}^{p} \) . | No |
Lemma 4. The upper envelope of a countable family of positive functions \( {f}_{n} \in {L}^{p} \) is in \( {L}^{p} \) if and only if there is function \( F \geq 0 \) such that \( {}^{11} \)\n\n\[ \n{N}_{p}\left( F\right) < + \infty \;\& \;{f}_{n}\left( x\right) \leq F\left( x\right) \text{ ae. for all }n.\n\] | The condition is necessary: for \( F \) take the upper envelope of the functions \( {f}_{n} \) . If it holds, the functions \( {g}_{n} = \sup \left( {{f}_{1},\ldots ,{f}_{n}}\right) \in {L}^{p} \) form an increasing sequence and satisfy \( {g}_{n} \leq F \) ae., whence \( {N}_{p}\left( {g}_{n}\right) \leq {N}_{p}\left(... | Yes |
Theorem 11 (Hölder). Let \( p, q > 1 \) and \( r \geq 1 \) be real numbers such that \( 1/p + 1/q = 1/r \) . Then\n\n(5.5)\n\n\[ f \in {L}^{p}\;\& \;g \in {L}^{q} \Rightarrow {fg} \in {L}^{r} \]\n\nand\n\n(5.6)\n\n\[ {N}_{r}\left( {fg}\right) \leq {N}_{p}\left( f\right) {N}_{q}\left( g\right) \] | Relation (6) is a particular case of (3.4'), which holds without any integrability assumptions. It, therefore, suffices to prove (5). To this end, we choose sequences \( \left( {f}_{n}\right) \) and \( \left( {g}_{n}\right) \) in \( L\left( X\right) \) such that\n\n\[ \lim {N}_{p}\left( {f - {f}_{n}}\right) = 0\;\& \;\... | Yes |
If \( f, g \in {L}^{2} \), then \( {fg} \) is integrable and\n\n\[ \left| {\mu \left( {fg}\right) }\right| \leq {N}_{2}\left( f\right) {N}_{2}\left( g\right) \] | This corollary shows that a Hilbert inner product\n\n\[ \left( {f \mid g}\right) = \mu \left( {f\bar{g}}\right) = \int f\left( x\right) \overline{g\left( x\right) }{d\mu }\left( x\right) \]\n\ncan be defined on \( {L}^{2} \). It is obviously linear \( {}^{14} \) in \( f \), satisfies the condition \( \left( {f \mid g}\... | No |
Corollary 2. If \( f \in {L}^{p} \), then \( f{\chi }_{K} \in {L}^{1} \) for every compact set \( K \subset X \) and \( {fg} \in {L}^{1} \) for all \( g \in L\left( X\right) \) . | Indeed, the function \( {\chi }_{K} \) is in \( {L}^{1}\left( {{\mathrm{n}}^{ \circ }4\text{, lemma 7}}\right) \) and hence is in \( {L}^{q} \) for all \( q \) (theorem 10), whence \( f{\chi }_{K} \in {L}^{1} \) . Use the same argument for the product \( {fg} \) . | No |
Lemma 5. The following two properties are equivalent for all functions \( \mathbf{j} \) :\n\n(LI 1) the function \( \mathbf{j}\left( x\right) {\chi }_{K}\left( x\right) \) is integrable for all compact sets \( K \) ;\n\n(LI 2) the function \( \mathbf{j}\left( x\right) f\left( x\right) \) is integrable for all \( f \in ... | (LI 1) \( \Rightarrow \) (LI 2): if \( f \) vanishes outside a compact set \( K \), then \( f\mathbf{j} = f \cdot \mathbf{j}{\chi }_{K} \) . Hence, corollary 2 applied to \( f{\chi }_{K} \) gives the result.\n\n(LI 2) \( \Rightarrow \) (LI 1): if \( f\mathbf{j} \in {L}^{1} \) for all \( f \in L\left( X\right), f \) can... | Yes |
Theorem 13. All functions belonging to a \( {L}^{p} \) space are measurable. | Because it is the limit almost everywhere of a sequence of continuous functions with values in \( \mathbb{C} \), the archetype of a metrizable and separable space. | No |
Theorem 14. Let \( P \) be a metrizable and separable set. The inverse image of every Borel set \( B \subset P \) under a measurable map \( f : X \rightarrow P \) is measurable. | Let \( \mathcal{T} \) be the set of subsets \( E \) of \( P \) such that \( {f}^{-1}\left( E\right) \) is measurable. Elementary formulae about the inverse image of an intersection or a complement show that, like the family of measurable subsets of \( X,\mathcal{T} \) is a tribe. However, by definition, \( \mathcal{T} ... | Yes |
Lemma 1. Let \( {f}_{1},\ldots ,{f}_{n} \) be maps from \( X \) to metrizable and separable sets \( {P}_{1},\ldots ,{P}_{n} \) . Then the map\n\n\[ f : x \mapsto \left( {{f}_{1}\left( x\right) ,\ldots ,{f}_{n}\left( x\right) }\right) \]\n\nfrom \( X \) to the product space \( P = {P}_{1} \times \ldots \times {P}_{n} \)... | First, the product space is metrizable and separable since if, for each \( i \) , \( {D}_{i} \) is everywhere dense in \( {P}_{i} \), then the product of the sets \( {D}_{i} \) is everywhere dense in \( P \) . If \( f \) is measurable, then, for any Borel set \( B \subset {P}_{1} \), the set\n\n\[ {f}^{-1}\left( {B \ti... | Yes |
Lemma 2. Let \( P \) and \( Q \) be two separable and metrizable spaces, \( \varphi \) a Borel map from \( P \) to \( Q \) and \( f \) a measurable map from \( X \) to \( P \) . Then \( g = \varphi \circ f \) : \( X \rightarrow Q \) is measurable. | It suffices to write that \( {g}^{-1}\left( B\right) = {f}^{-1}\left( {B}^{\prime }\right) \), where \( {B}^{\prime } = {\varphi }^{-1}\left( B\right) \). | Yes |
Theorem 15. Let \( {P}_{1},\ldots ,{P}_{n} \) and \( Q \) be separable and metrizable spaces, \( \varphi \) a Borel map from \( {P}_{1} \times \ldots \times {P}_{n} \) to \( Q \) and \( {f}_{i} : X \rightarrow {P}_{i} \) measurable maps. Then the map\n\n\[ x \mapsto \varphi \left\lbrack {{f}_{1}\left( x\right) ,\ldots ... | This is a direct consequence of the previous two lemma. | No |
Theorem 17 (Lusin). A function \( f \) with values in a metrizable and separable space \( P \) is measurable if and only if \( f \) satisfies condition (LUS). | (a) Let us first suppose that there is a countable partition of \( X \) into measurable sets \( {E}_{n} \) such that \( f \) is constant on each of them. We will then say that \( f \) is a step function. This is a less trivial analogue of the step functions of Chap. V, \( {\mathrm{n}}^{ \circ } 1. All functions of this... | No |
Theorem 18. A function \( f \) with values \( i{n}^{25}\mathbb{C} \) or \( \left\lbrack {-\infty , + \infty }\right\rbrack \) is in \( {L}^{p} \) if and only if it is measurable and \( {N}_{p}\left( f\right) < + \infty \) . | (a) As \( \left| {f\left( x\right) }\right| < + \infty \) ae., we need only consider the case of finite-valued function. By Lusin, there is a sequence of compact sets \( {K}_{n} \), which may be assumed to be increasing, such that \( f \) is continuous on each \( {K}_{n} \) and zero ae. outside the union of these \( {K... | Yes |
Corollary 1. The product of a function of \( {L}^{p} \) and a bounded measurable function is in \( {L}^{p} \) . | Obvious. | No |
Corollary 3. Let \( f \) be a reasonable and measurable function with values in \( \left\lbrack {0, + \infty }\right\rbrack \) . Then there is an increasing sequence \( \left( {f}_{n}\right) \) of bounded positive integrable functions such that\n\n\[ f\left( x\right) = \lim {f}_{n}\left( x\right) \;\text{ for all }x \i... | Indeed, let \( {A}_{n} \) be an increasing sequence of integrable sets whose union is the measurable set \( \{ f\left( x\right) \neq 0\} \) . We set\n\n\[ {f}_{n}\left( x\right) = \begin{array}{l} \inf \left\lbrack {f\left( x\right), n}\right\rbrack \;\text{ if }x \in {A}_{n}, \\ 0\;\text{ otherwise. } \end{array} \]\n... | Yes |
Lemma 2. A locally compact metric space \( X \) is separable if and only if it is countable at infinity. | Indeed if \( X \) is the countable union of compact sets \( {K}_{n} \), each \( {K}_{n} \) contains an everywhere dense countable subset \( {D}_{n} \) . This gives the result for \( X \) by considering the union of all \( {D}_{n} \) . Conversely, if \( \left( {U}_{n}\right) \) is a countable basis for the topology on \... | Yes |
Lemma 3. Every open subspace \( P \) of a Polish space \( X \) is Polish. | Let \( P = X - F \), where \( F \) is closed, and we set\n\n\[ f\left( x\right) = d\left( {x, F}\right) \]\n\nassuming that \( X \) is complete with respect to \( d \) . The relation \( f\left( x\right) \neq 0 \) is equivalent to \( x \in P \) . In the Cartesian product \( \mathbb{R} \times X \), which is obviously\n\n... | Yes |
Corollary 1. Let \( P \) a metrizable space. Suppose that \( P \) is complete with respect to some distance compatible with its topology. Then \( P \) is a \( {G}_{\delta } \) set in its completion with respect to any distance compatible with its topology. | Obvious. | No |
Every Polish space is homeomorphic to a \( {G}_{\delta } \) set in a compact Polish space. | To prove this, let us consider the infinite-dimensional cube \( {I}^{\mathbb{N}} \), where \( I = \left\lbrack {0,1}\right\rbrack \) . It is the set of maps or sequences \( u : \mathbb{N} \rightarrow I \) equipped with the distance\n\n\[ d\left( {u, v}\right) = \sum {2}^{-n}\left| {{u}_{n} - {v}_{n}}\right| \]\n\nwhich... | No |
Theorem 22. Let \( X \) and \( Z \) be locally compact spaces, \( {\mu }_{x} \) a family of positive measures on \( Z \) depending on a parameter \( x \in X \), and \( \lambda \) a positive measure on \( X \) . Suppose that condition (INT) holds and that this is also the case of one of the following conditions:\n\n( \(... | Proposition (i) in the statement is obvious. \( {}^{44} \) | No |
Theorem 23. Let \( X \) and \( Z \) be two locally compact spaces, \( \lambda \) a positive measure on \( X \) and \( p \) a measurable map from \( X \) to \( Z \) . Suppose that, for all \( f \in L\left( Z\right) \), the function \( f \circ p \) is \( \lambda \) -integrable. Let\n\n\[ \nu \left( f\right) = \lambda \le... | (i) A reasonable function \( f \) on \( Z \) is \( \nu \) -integrable if and only if \( f \circ p \) is \( \lambda \) -integrable; then\n\n(14.3)\n\n\[ \nu \left( f\right) = \lambda \left( {f \circ p}\right) .\] | Yes |
Lemma 1. Let \( X \) and \( Z \) be two locally compact spaces and \( p \) a continuous proper map from \( X \) to \( Z \) . The image \( p\left( F\right) \) of every closed set \( F \subset X \) is closed in \( Z \) . | The restriction of \( p \) to \( F \) being proper, we may assume that \( X = F \) . Let \( z \in Z \) be a closure point of \( p\left( X\right) \) . Since every compact neighbourhood \( W \) of \( z \) has non-trivial intersection with \( p\left( X\right) \), the compact sets \( {p}^{-1}\left( W\right) \) are non-empt... | No |
Lemma 2. Let \( X \) and \( Z \) be two locally compact spaces, \( p \) a continuous proper surjective map from \( X \) to \( Z \) and \( f \) a map from \( Z \) to a topological space. \( f \) is continuous if (and only if) \( f \circ p \) is continuous. | By lemma \( 1, p \) maps every closed subset of \( X \) onto a closed subset of \( Z \) . Hence for a set \( A \subset Z \) to be closed it is sufficient (and necessary since \( p \) is continuous) for \( {p}^{-1}\left( A\right) \) to be closed. Hence \( V \subset Z \) is open if and only if \( {p}^{-1}\left( V\right) ... | Yes |
Lemma 1. Let \( Z \) be a separated topological space and \( R \subset Z \times Z \) the graph \( {}^{56} \) of an equivalence relation on \( Z \) such that the map \( p : Z \mapsto Z/R \) is open. \( Z/R \) is separated if and only if \( R \) is closed in \( Z \times Z \) . | Let us set \( {Z}^{\prime } = Z/R \) . The map \( \left( {x, y}\right) \mapsto \left( {p\left( x\right), p\left( y\right) }\right) \) from \( Z \times Z \) to \( {Z}^{\prime } \times {Z}^{\prime } \) is continuous. However, the set \( R \) of ordered pairs such that \( p\left( x\right) = p\left( y\right) \) is the inve... | No |
Lemma 2. Let \( G \) be a group acting on a separated topological space \( Z \) . The quotient space \( G \smallsetminus Z \) is separated if and only if the following condition holds: for every pair of points \( c,{c}^{\prime } \in Z \) such that \( {Gc} \neq G{c}^{\prime } \), there are neighbourhoods \( U \) and \( ... | We denote by \( R \) the equivalence relation defined by \( G \) . As was seen above, the map \( p \) is open. Let \( \left( {c,{c}^{\prime }}\right) \) be a closure point of \( R \) . Any neighbourhood of \( \left( {c,{c}^{\prime }}\right) \) contains a set \( U \times {U}^{\prime } \), where \( U \) and \( {U}^{\prim... | No |
Lemma 4. The map \( f \mapsto {f}^{G} \) from \( {L}_{ + }\left( Z\right) \) to \( {L}_{ + }\left( X\right) \) is surjective. | First note that any compact set \( K \subset X \) is the image under \( p \) of a compact subset of \( Z \) . For all \( x = p\left( z\right) \) and all compact neighbourhoods \( W \) of \( z \), the image \( p\left( W\right) \) is indeed a compact neighbourhood of \( x \) . By BL, \( K \) can be covered by finitely ma... | Yes |
Lemma 6. Let \( \Gamma \) be a discrete group acting on a locally compact space \( Z \) . The quotient space \( \Gamma \smallsetminus Z \) is locally compact, if the following condition holds:\n\n(GPD) for all compact sets \( A, B \subset Z \), the set of \( \gamma \in \Gamma \) such that \( {\gamma A}\# B \) is finite... | If \( z,{z}^{\prime } \) are two points of \( Z \) and \( U,{U}^{\prime } \) are sufficiently small neighbourhoods of \( z \) and \( {z}^{\prime } \), then\n\n(15.19)\n\n\[ {\gamma U}\# {U}^{\prime } \Leftrightarrow {\gamma z} = {z}^{\prime } \]\n\nThe first proposition is clear. If \( {\Gamma z} \neq \Gamma {z}^{\prim... | Yes |
Theorem 25. Let \( X \) be a locally compact Polish space, \( \lambda \) a positive measure on \( X \) and \( \mathbf{j} \geq 0 \) a locally \( \lambda \) -integrable function. A function \( f \) with values in \( \mathbb{C} \) or \( \left\lbrack {-\infty , + \infty }\right\rbrack \) is integrable with respect to the m... | (a) As observed at the end of section (ii) of \( {\mathrm{n}}^{ \circ }{12} \), it suffices to set \( Z = X \) and\n\n(16.4)\n\n\[{\mu }_{x}\left( f\right) = \mathbf{j}\left( x\right) f\left( x\right) \;\text{ for all }f \in L\left( X\right)\]\nin order to put \( \nu \) into the form\n\n(16.5)\n\n\[\nu \left( f\right) ... | Yes |
Lemma 2. Let \( f \) be a function with values in \( \mathbb{C} \) or \( \left\lbrack {-\infty , + \infty }\right\rbrack \) . Then \( f \) is measurable with respect to \( {d\nu } = \mathbf{j}{d\lambda } \) if and only if \( f\mathbf{j} \) is \( \lambda \) -measurable. | The function \( {\mathbf{j}}^{\prime }\left( x\right) \) equal to \( 1/\mathbf{j}\left( x\right) \) on \( S \) and 0 on \( X - S \), being \( \lambda \) -measurable like \( \mathbf{j} \) and constant outside \( S \), is \( \nu \) -measurable by theorem 26. So, if \( f\mathbf{j} \) is \( \lambda \) - measurable, and hen... | Yes |
Theorem 27. Let \( \lambda \) and \( \nu \) be two positive measures on a locally compact Polish space \( X{.}^{68} \) There is a locally \( \lambda \) -integrable function \( \mathbf{j} \) such that \( {d\nu }\left( x\right) = \) \( \mathbf{j}\left( x\right) {d\lambda }\left( x\right) \) if and only if every \( \lambd... | (a) Let us first suppose that \( \nu \leq \lambda \) and that there is a compact set \( K \) such that \( \nu \left( {X - K}\right) = 0 \) . The first assumption shows that \( {\nu }^{ * }\left( f\right) \leq {\lambda }^{ * }\left( f\right) \) for any function \( f \geq 0 \) . Then, for all \( f \in L\left( X\right) \)... | No |
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