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Lemma 2 Let \( f : \mathbb{R} \rightarrow \mathbb{C} \) be a continuous function having a locally piecewise continuous derivative \( {f}^{\prime } \) on \( \mathbb{R} \) . Given this,\n\na) if the function \( {f}^{\prime } \) is integrable on \( \mathbb{R} \), then \( f\left( x\right) \) has a limit both as \( x \right... | Proof Under these restrictions on the functions \( f \) and \( {f}^{\prime } \) the Newton-Leibniz formula holds\n\n\[ f\left( x\right) = f\left( 0\right) + {\int }_{0}^{x}{f}^{\prime }\left( t\right) \mathrm{d}t. \]\n\nIn conditions a) the right-hand side of this equality has a limit both as \( x \rightarrow + \infty ... | Yes |
Proposition 1 (Connection between the smoothness of a function and the rate of decrease of its Fourier transform) If \( f \in {C}^{\left( k\right) }\left( {\mathbb{R},\mathbb{C}}\right) \left( {k = 0,1,\ldots }\right) \) and all the functions \( f,{f}^{\prime },\ldots ,{f}^{\left( k\right) } \) are absolutely integrabl... | Proof If \( k = 0 \), then a) holds trivially and b) follows from the Riemann-Lebesgue lemma.\n\nLet \( k > 0 \) . By Lemma 2 the functions \( f,{f}^{\prime },\ldots ,{f}^{\left( k - 1\right) } \) tend to zero as \( x \rightarrow \infty \) . Taking this into account, we integrate by parts,\n\n\[ \widehat{{f}^{\left( k\... | Yes |
Proposition 2 (The connection between the rate of decrease of a function and the smoothness of its Fourier transform) If a locally integrable function \( f : \mathbb{R} \rightarrow \mathbb{C} \) is such that the function \( {x}^{k}f\left( x\right) \) is absolutely integrable on \( \mathbb{R} \), then\na) the Fourier tr... | Proof For \( k = 0 \) relation (18.103) holds trivially, and the continuity of \( \widehat{f}\left( \xi \right) \) has already been proved in Lemma 1. If \( k > 0 \), then for \( n < k \) we have the estimate \( \left| {{x}^{n}f\left( x\right) }\right| \leq \left| {{x}^{k}f\left( x\right) }\right| \) at infinity, from ... | Yes |
Lemma 3 The restriction of the Fourier transform to \( S \) is a vector-space automorphism of \( S \) . | Proof We first show that \( \left( {f \in S}\right) \Rightarrow \left( {\widehat{f} \in S}\right) \) .\n\nTo do this we first remark that by Proposition 2a we have \( \widehat{f} \in {C}^{\left( \infty \right) }\left( {\mathbb{R},\mathbb{C}}\right) \) .\n\nWe then remark that the operation of multiplication by \( {x}^{... | Yes |
The function \( {\mathrm{e}}^{-{\left| x\right| }^{2}} \), where \( {\left| x\right| }^{2} = {x}_{1}^{2} + \cdots + {x}_{n}^{2} \), and all the functions in \( {C}_{0}^{\left( \infty \right) }\left( {{\mathbb{R}}^{n},\mathbb{C}}\right) \) of compact support belong to \( S \) . | If \( f \in S \), then integral in relation (18.104) obviously converges absolutely and uniformly with respect to \( \xi \) on the entire space \( {\mathbb{R}}^{n} \) . Moreover, if \( f \in S \), then by standard rules this integral can be differentiated as many times as desired with respect to any of the variables \(... | No |
Let us find the Fourier transform of the function \( \exp \left( {-{\left| x\right| }^{2}/2}\right) \) . | In the present case, using Fubini's theorem and Example 4, we find\n\n\[ \frac{1}{{\left( 2\pi \right) }^{n/2}}{\int }_{{\mathbb{R}}^{n}}{\mathrm{e}}^{-{\left| x\right| }^{2}/2} \cdot {\mathrm{e}}^{-i\left( {\xi, x}\right) }\mathrm{d}x = \]\n\n\[ = \mathop{\prod }\limits_{{j = 1}}^{n}\frac{1}{\sqrt{2\pi }}{\int }_{-\in... | Yes |
In describing relative motions occurring with speeds \( v \) that are much smaller than the speed of light \( \left( {\left| v\right| \ll c}\right) \) we may use, instead of the Lorentz transformations (Example 3 of Sect. 1.3) | \[ {x}^{\prime } = \frac{x - {vt}}{\sqrt{1 - {\left( \frac{v}{c}\right) }^{2}}},\;{t}^{\prime } = \frac{t - \left( \frac{v}{{c}^{2}}\right) x}{\sqrt{1 - {\left( \frac{v}{c}\right) }^{2}}}, \] the Galilean transformation \[ {x}^{\prime } = x - {vt},\;{t}^{\prime } = t, \] since \( v/c \approx 0 \) . | Yes |
Example 2 The period\n\n\[ T = 4\sqrt{\frac{l}{g}}{\int }_{0}^{\pi /2}\frac{\mathrm{d}\theta }{\sqrt{1 - {k}^{2}{\sin }^{2}\theta }} \]\n\nof oscillations of a pendulum is connected with the maximal angle of deviation \( {\varphi }_{0} \) from its equilibrium position via the parameter \( {k}^{2} = {\sin }^{2}\frac{{\v... | \[ T \approx {2\pi }\sqrt{\frac{l}{g}} \] | No |
The labor involved in computing the values of \( n \) ! or \( \ln n \) ! increase as \( n \in \mathbb{N} \) increases. We shall use the fact that \( n \) is large, however, and obtain under that assumption a convenient asymptotic formula for computing \( \ln n \) ! approximately. | It follows from the obvious relations\n\n\[ \n{\int }_{1}^{n}\ln x\mathrm{\;d}x = \mathop{\sum }\limits_{{k = 2}}^{n}{\int }_{k - 1}^{k}\ln x\mathrm{\;d}x < \mathop{\sum }\limits_{{k = 1}}^{n}\ln k < \mathop{\sum }\limits_{{k = 2}}^{n}{\int }_{k}^{k + 1}\ln x\mathrm{\;d}x = {\int }_{2}^{n + 1}\ln x\mathrm{\;d}x \]\n\nt... | Yes |
Example 7 We shall show that as \( x \rightarrow + \infty \) the function\n\n\[ \n{f}_{n}\left( x\right) = {\int }_{1}^{x}\frac{{\mathrm{e}}^{t}}{{t}^{n}}\mathrm{\;d}t\;\left( {n \in \mathbb{R}}\right) \n\]\n\nis asymptotically equivalent to the function \( {g}_{n}\left( x\right) = {x}^{-n}{\mathrm{e}}^{x} \) . | Since \( {g}_{n}\left( x\right) \rightarrow + \infty \) as \( x \rightarrow + \infty \), applying L’Hôpital’s rule we find\n\n\[ \n\mathop{\lim }\limits_{{x \rightarrow + \infty }}\frac{{f}_{n}\left( x\right) }{{g}_{n}\left( x\right) } = \mathop{\lim }\limits_{{x \rightarrow + \infty }}\frac{{f}_{n}^{\prime }\left( x\r... | Yes |
Let us find the asymptotic behavior of the function\n\n\\[ f\\left( x\\right) = {\\int }_{1}^{x}\\frac{{\\mathrm{e}}^{t}}{t}\\mathrm{\\;d}t \\]\n\nmore precisely. | Integrating by parts, we obtain\n\n\\[ f\\left( x\\right) = {\\left. \\frac{{e}^{t}}{t}\\right| }_{1}^{x} + {\\int }_{1}^{x}\\frac{{\\mathrm{e}}^{t}}{{t}^{2}}\\mathrm{\\;d}t = {\\left. \\left( \\frac{{e}^{t}}{t} + \\frac{{e}^{t}}{{t}^{2}}\\right) \\right| }_{1}^{x} + {\\int }_{1}^{x}\\frac{2{\\mathrm{e}}^{t}}{{t}^{3}}\... | Yes |
Proposition 3 (Integration of asymptotic equalities) Let \( f \) be a continuous function on the interval \( I = \lbrack a,\omega \left\lbrack {(\text{or}I = }\right\rbrack \omega, a\rbrack ) \). a) If the function \( g\left( x\right) \) is continuous and nonnegative on 1 and the integral \( {\int }_{a}^{\omega }g\left... | Proof a) If \( f\left( x\right) = O\left( {g\left( x\right) }\right) \) as \( I \ni x \rightarrow \omega \), there exists \( {x}_{0} \in I \) and a constant \( M \) such that \( \left| {f\left( x\right) }\right| \leq {Mg}\left( x\right) \) for \( x \in \left\lbrack {{x}_{0},\omega \lbrack }\right. \) . It follows that ... | Yes |
The function \( f\left( x\right) = {\mathrm{e}}^{-x}\sin \left( {\mathrm{e}}^{x}\right) \) is continuously differentiable on \( \mathbb{R} \) and is an asymptotic zero with respect to the asymptotic sequence \( \left\{ \frac{1}{{x}^{n}}\right\} \) as \( x \rightarrow + \infty \). | The derivatives of the functions \( \frac{1}{{x}^{n}} \), up to a constant factor, again have the form \( \frac{1}{{x}^{k}} \) . However the function \( {f}^{\prime }\left( x\right) = - {\mathrm{e}}^{x}\sin \left( {\mathrm{e}}^{x}\right) + \cos \left( {\mathrm{e}}^{x}\right) \) not only fails to be an asymptotic zero; ... | Yes |
Proposition 4 Let 0 be a limit point of \( E \) and let\n\n\[ f\left( x\right) \simeq {a}_{0} + {a}_{1}x + {a}_{2}{x}^{2} + \cdots ,\]\n\n\[ g\left( x\right) \simeq {b}_{0} + {b}_{1}x + {b}_{2}{x}^{2} + \cdots \]\n\nThen as \( E \ni x \rightarrow 0 \),\n\na) \( \left( {{\alpha f} + {\beta g}}\right) \simeq \mathop{\sum... | Proof a) This is a special case of Proposition 2.\n\nb) Using the properties of \( o\left( \right) \) (see Proposition 4 of Sect. 3.2), we find that\n\n\[ \left( {f \cdot g}\right) \left( x\right) = \]\n\n\[ = f\left( x\right) \cdot g\left( x\right) = \]\n\n\[ = \left( {{a}_{0} + {a}_{1}x + \cdots + {a}_{n}{x}^{n} + o\... | Yes |
If \( U \) is a neighborhood (or one-sided neighborhood) of infinity in \( \mathbb{R} \) and the function \( f \) is continuous in \( U \) and has the asymptotic expansion\n\n\[ f\left( x\right) \simeq {a}_{0} + \frac{{a}_{1}}{x} + \frac{{a}_{2}}{{x}^{2}} + \cdots + \frac{{a}_{n}}{{x}^{n}} + \cdots \;\text{ as }U \ni x... | Proof The convergence of the integral is obvious, since\n\n\[ f\left( t\right) - {a}_{0} - \frac{{a}_{1}}{t} \sim \frac{{a}_{2}}{{t}^{2}}\;\text{ as }U \ni t \rightarrow \infty . \]\n\nIt remains only to integrate the asymptotic expansion\n\n\[ f\left( t\right) - {a}_{0} - \frac{{a}_{1}}{t} \simeq \frac{{a}_{2}}{{t}^{2... | Yes |
Corollary 2 If in addition to the hypotheses of Corollary 1 it is known that \( f \in \) \( {C}^{\left( 1\right) }\left( U\right) \) and \( {f}^{\prime } \) admits the asymptotic expansion\n\n\[ \n{f}^{\prime }\left( x\right) \simeq {a}_{0}^{\prime } + \frac{{a}_{1}^{\prime }}{x} + \frac{{a}_{2}^{\prime }}{{x}^{2}} + \... | Proof Since \( {f}^{\prime }\left( x\right) = {a}_{0}^{\prime } + \frac{{a}_{1}^{\prime }}{x} + O\left( {1/{x}^{2}}\right) \) as \( U \ni x \rightarrow \infty \), we have\n\n\[ \nf\left( x\right) = f\left( {x}_{0}\right) + {\int }_{{x}_{0}}^{x}{f}^{\prime }\left( t\right) \mathrm{d}t = {a}_{0}^{\prime }x + {a}_{1}^{\pr... | Yes |
Example 2 Laplace himself applied his method to integrals of the form \( {\int }_{a}^{b}f\left( x\right) {\varphi }^{n}\left( x\right) \mathrm{d}x \), where \( n \in \mathbb{N} \) and \( \varphi \left( x\right) > 0 \) on \( \rbrack a, b\lbrack \) . Such an integral is also a special case of a general Laplace integral (... | We shall be interested in the asymptotics of the integral (19.1) for large values of the parameter \( \lambda \), more precisely as \( \lambda \rightarrow + \infty ,\lambda \in \mathbb{R} \) . So as not to become distracted with secondary issues when describing the basic idea of Laplace’s method, we shall assume that \... | "No" |
Example 3 Let \( {x}_{0} = a,{S}^{\prime }\left( a\right) \neq 0 \), and \( f\left( a\right) \neq 0 \), which happens, for example, when the function \( S\left( x\right) \) is monotonically decreasing on \( \left\lbrack {a, b}\right\rbrack \) . Under these conditions \( f\left( x\right) = f\left( a\right) + o\left( 1\r... | Carrying out the idea of Laplace’s method, for a small \( \varepsilon > 0 \) and \( \lambda \rightarrow + \infty \), we find that\n\n\[ F\left( \lambda \right) \sim {\int }_{a}^{a + \varepsilon }f\left( x\right) {\mathrm{e}}^{{\lambda S}\left( x\right) }\mathrm{d}x \sim \]\n\n\[ \sim f\left( a\right) {\mathrm{e}}^{{\la... | Yes |
Example 4 Let \( a < {x}_{0} < b \) . Then \( {S}^{\prime }\left( {x}_{0}\right) = 0 \), and we assume that \( {S}^{\prime \prime }\left( {x}_{0}\right) \neq 0 \), that is, \( {S}^{\prime \prime }\left( {x}_{0}\right) < 0 \), since \( {x}_{0} \) is a maximum. | Using the expansions \( f\left( x\right) = f\left( {x}_{0}\right) + o\left( {x - {x}_{0}}\right) \) and \( S\left( x\right) = S\left( {x}_{0}\right) + \frac{1}{2}{S}^{\prime \prime }\left( {x}_{0}\right) (x - \) \( {\left. {x}_{0}\right) }^{2} + o\left( {\left( x - {x}_{0}\right) }^{2}\right) \), which hold as \( x \ri... | Yes |
If \( {x}_{0} = a \), but \( {S}^{\prime }\left( {x}_{0}\right) = 0 \) and \( {S}^{\prime \prime }\left( {x}_{0}\right) < 0 \), then, reasoning as in Example 4 , we find this time that | \[ F\left( \lambda \right) \sim {\int }_{a}^{a + \varepsilon }f\left( x\right) {\mathrm{e}}^{{\lambda S}\left( x\right) }\mathrm{d}x \sim f\left( {x}_{0}\right) {\mathrm{e}}^{{\lambda S}\left( {x}_{0}\right) }{\int }_{0}^{\varepsilon }{\mathrm{e}}^{\frac{1}{2}\lambda {S}^{\prime \prime }\left( {x}_{0}\right) {t}^{2}}\m... | Yes |
Lemma 1 (Exponential estimate) Let \( M = \mathop{\sup }\limits_{{a < x < b}}S\left( x\right) < \infty \), and suppose that for some value \( {\lambda }_{0} > 0 \) the integral (19.1) converges absolutely. Then it converges absolutely for every \( \lambda \geq {\lambda }_{0} \) and the following estimate holds for such... | Proof Indeed, for \( \lambda \geq {\lambda }_{0} \) ,\n\n\[ \left| {F\left( \lambda \right) }\right| = \left| {{\int }_{a}^{b}f\left( x\right) {\mathrm{e}}^{{\lambda S}\left( x\right) }\mathrm{d}x}\right| = \left| {{\int }_{a}^{b}f\left( x\right) {\mathrm{e}}^{{\lambda }_{0}S\left( x\right) }{\mathrm{e}}^{\left( {\lamb... | Yes |
Lemma 2 (Estimate of the contribution of a maximum point) Suppose the integral (19.1) converges absolutely for some value \( \lambda = {\lambda }_{0} \), and suppose that in the interior or on the boundary of the interval \( I \) there is a point \( {x}_{0} \) at which \( S\left( {x}_{0}\right) = \mathop{\sup }\limits_... | Proof For a fixed \( \varepsilon > 0 \) let us take any neighborhood \( {U}_{I}\left( {x}_{0}\right) \) inside which \( \left| {f\left( x\right) }\right| \geq \) \( \frac{1}{2}\left| {f\left( {x}_{0}\right) }\right| \) and \( S\left( {x}_{0}\right) - \varepsilon \leq S\left( x\right) \leq S\left( {x}_{0}\right) \) . As... | Yes |
Proposition 1 (Localization principle) Suppose the integral (19.1) converges absolutely for a value \( \lambda = {\lambda }_{0} \), and suppose that inside or on the boundary of the interval I of integration the function \( S\left( x\right) \) has a unique point \( {x}_{0} \) of absolute maximum, that is, outside every... | Proof It follows from Lemma 2 that if the neighborhood \( {U}_{I}\left( {x}_{0}\right) \) is sufficiently small, then the following inequality holds ultimately as \( \lambda \rightarrow + \infty \) for every \( \varepsilon > 0 \)\n\n\[ \left| {{F}_{{U}_{I}\left( {x}_{0}\right) }\left( \lambda \right) }\right| > {\mathr... | Yes |
Lemma 3 (Canonical form of the function in the neighborhood of a critical point) If the real-valued function \( S\left( x\right) \) has smoothness \( {C}^{\left( n + k\right) } \) in a neighborhood (or one-sided neighborhood) of a point \( {x}_{0} \in \mathbb{R} \), and\n\n\[ \n{S}^{\prime }\left( {x}_{0}\right) = \cdo... | Proof Using Taylor's formula with the integral form of the remainder,\n\n\[ \nS\left( x\right) = S\left( {x}_{0}\right) + \frac{{\left( x - {x}_{0}\right) }^{n}}{\left( {n - 1}\right) !}{\int }_{0}^{1}{S}^{\left( n\right) }\left( {{x}_{0} + t\left( {x - {x}_{0}}\right) }\right) {\left( 1 - t\right) }^{n - 1}\mathrm{\;d... | Yes |
Proposition 2 (Reduction) Suppose the interval of integration \( I = \left\lbrack {a, b}\right\rbrack \) in the integral (19.1) is finite and the following conditions hold:\n\na) \( f, S \in C\left( {I,\mathbb{R}}\right) \) ;\n\nb) \( \mathop{\max }\limits_{{x \in I}}S\left( x\right) \) is attained only at the one poin... | Proof Using the localization principle, we replace the integral (19.1) with the integral over a neighborhood \( {I}_{x} = {U}_{I}\left( {x}_{0}\right) \) of \( {x}_{0} \) in which the hypotheses of Lemma 3 hold. Making the change of variable \( x = \varphi \left( y\right) \), we obtain\n\n\[ {\int }_{{I}_{x}}f\left( x\... | Yes |
Lemma 4 (Watson \( {}^{3} \) ) Let \( \alpha > 0,\beta > 0,0 < a \leq \infty \), and \( f \in C\left( {\left\lbrack {0, a}\right\rbrack ,\mathbb{R}}\right) \) . Then with respect to the asymptotics of the integral\n\n\[ W\left( \lambda \right) = {\int }_{0}^{a}{x}^{\beta - 1}f\left( x\right) {\mathrm{e}}^{-\lambda {x}^... | Proof We represent the integral (19.12) as a sum of integrals over the interval \( \rbrack 0,\varepsilon \rbrack \) and \( \lbrack \varepsilon, a\lbrack \), where \( \varepsilon \) is an arbitrarily small positive number.\n\nBy Lemma 1\n\n\[ \left| {{\int }_{\varepsilon }^{a}{x}^{\beta - 1}f\left( x\right) {\mathrm{e}}... | Yes |
Consider the Laplace transform\n\n\[ \nF\left( \lambda \right) = {\int }_{0}^{+\infty }f\left( x\right) {\mathrm{e}}^{-{\lambda x}}\mathrm{\;d}x \]\n\nwhich we have already encountered in Example 1. If this integral converges absolutely for some value \( \lambda = {\lambda }_{0} \) and the function \( f \) is infinitel... | \[ \nF\left( \lambda \right) \simeq \mathop{\sum }\limits_{{k = 0}}^{\infty }{f}^{\left( k\right) }\left( 0\right) {\lambda }^{-\left( {k + 1}\right) }\;\text{ as }\lambda \rightarrow + \infty . \] | Yes |
Theorem 1 (A typical principal term of the asymptotics) Suppose the interval of integration \( I = \left\lbrack {a, b}\right\rbrack \) in the integral (19.1) is finite, \( f, S \in C\left( {I,\mathbb{R}}\right) \), and \( \mathop{\max }\limits_{{x \in I}}S\left( x\right) \) is attained only at one point \( {x}_{0} \in ... | Proof Using the localization principle and making the change of variable \( x = \varphi \left( y\right) \) shown in Lemma 3, according to the reduction in Proposition 2, we arrive at the following relations: \[ \text{a)}F\left( \lambda \right) = {\mathrm{e}}^{{\lambda S}\left( {x}_{0}\right) }\left( {{\int }_{0}^{\vare... | Yes |
The asymptotics of the gamma function. The function\n\n\[ \n\Gamma \left( {\lambda + 1}\right) = {\int }_{0}^{+\infty }{t}^{\lambda }{\mathrm{e}}^{-t}\mathrm{\;d}t\;\left( {\lambda > - 1}\right) \n\] | can be represented as a Laplace integral\n\n\[ \n\Gamma \left( {\lambda + 1}\right) = {\int }_{0}^{+\infty }{\mathrm{e}}^{-t}{\mathrm{e}}^{\lambda \ln t}\mathrm{\;d}t \n\]\n\nand if for \( \lambda > 0 \) we make the change of variable \( t = {\lambda x} \), we arrive at the integral\n\n\[ \n\Gamma \left( {\lambda + 1}\... | Yes |
The asymptotics of the Bessel function\n\n\[ \n{I}_{n}\left( x\right) = \frac{1}{\pi }{\int }_{0}^{\pi }{\mathrm{e}}^{x\cos \theta }\cos {n\theta }\mathrm{d}\theta , \n\]\n\nwhere \( n \in \mathbb{N} \) . | Here \( f\left( \theta \right) = \cos {n\theta }, S\left( \theta \right) = \cos \theta ,\mathop{\max }\limits_{{0 \leq x \leq \pi }}S\left( \theta \right) = S\left( 0\right) = 1 \) , \( {S}^{\prime }\left( 0\right) = 0 \), and \( {S}^{\prime \prime }\left( 0\right) = - 1 \), so that by assertion c) of Theorem 1\n\n\[ \... | Yes |
Let \( f \in {C}^{\left( 1\right) }\left( {\left\lbrack {a, b}\right\rbrack ,\mathbb{R}}\right), S \in {C}^{\left( 2\right) }\left( {\left\lbrack {a, b}\right\rbrack ,\mathbb{R}}\right) \), with \( S\left( x\right) > 0 \) on \( \left\lbrack {a, b}\right\rbrack \) , and \( \mathop{\max }\limits_{{a \leq x \leq b}}S\left... | on the basis of assertions b) and c) of Theorem 1, we find that as \( \lambda \rightarrow + \infty \)\n\n\[ \mathcal{F}\left( \lambda \right) = {\varepsilon f}\left( {x}_{0}\right) \sqrt{\frac{2\pi }{-{S}^{\prime \prime }\left( {x}_{0}\right) }}{\left\lbrack S\left( {x}_{0}\right) \right\rbrack }^{\lambda + 1/2}{\lambd... | Yes |
The asymptotics of the Legendre polynomials\n\n\[ \n{P}_{n}\left( x\right) = \frac{1}{\pi }{\int }_{0}^{\pi }{\left( x + \sqrt{{x}^{2} - 1}\cos \theta \right) }^{n}\mathrm{\;d}\theta \n\]\n\nin the domain \( x > 1 \) as \( n \rightarrow \infty, n \in \mathbb{N} \), can be obtained as a special case of the preceding exa... | \n\[ \nS\left( \theta \right) = x + \sqrt{{x}^{2} - 1}\cos \theta ,\;\mathop{\max }\limits_{{0 \leq \theta \leq \pi }}S\left( \theta \right) = S\left( 0\right) = x + \sqrt{{x}^{2} - 1}, \n\]\n\n\[ \n{S}^{\prime }\left( 0\right) = 0,\;{S}^{\prime \prime }\left( 0\right) = - \sqrt{{x}^{2} - 1}. \n\]\n\nThus,\n\n\[ \n{P}_... | Yes |
Theorem 2 (Asymptotic expansion) Let \( I = \left\lbrack {a, b}\right\rbrack \) be a finite interval, \( f, S \in \) \( C\left( {I,\mathbb{R}}\right) \), and assume \( \mathop{\max }\limits_{{x \in I}}S\left( x\right) \) is attained only at the point \( {x}_{0} \in I \) and \( f, S \in \) \( {C}^{\left( \infty \right) ... | Proof It follows from Lemma 1 that under these hypotheses the integral (19.1) can be replaced by an integral over an arbitrarily small neighborhood of \( {x}_{0} \) up to a quantity of the form \( {\mathrm{e}}^{\lambda \widetilde{S}\left( {x}_{0}\right) }O\left( {\lambda }^{-\infty }\right) \) as \( \lambda \rightarrow... | Yes |
If \( S \in {C}^{\left( \infty \right) }\left( {\mathbb{R},\mathbb{R}}\right) \) and \( S\left( x\right) \rightarrow - \infty \) as \( x \rightarrow \infty \), then \[ F\left( \lambda \right) = {\int }_{-\infty }^{\infty }{S}^{\prime }\left( x\right) {\mathrm{e}}^{{\lambda S}\left( x\right) }\mathrm{d}x \equiv 0\;\text... | Hence, in this case such an interference of the contributions must necessarily occur. From the formal point of view this example may seem unconvincing, since previously we had been considering the case of a finite interval of integration. However, those doubts are removed by the following important remark. | No |
The asymptotic behavior of the function\n\n\[ \operatorname{Erf}\left( x\right) = {\int }_{x}^{+\infty }{\mathrm{e}}^{-{u}^{2}}\mathrm{\;d}u \] \n\nas \( x \rightarrow + \infty \) | is easy to obtain through integration by parts:\n\n\[ \operatorname{Erf}\left( x\right) = \frac{{\mathrm{e}}^{-{x}^{2}}}{2x} - \frac{1}{2}{\int }_{x}^{+\infty }{u}^{-2}{\mathrm{e}}^{-{u}^{2}}\mathrm{\;d}u = \frac{{e}^{-{x}^{2}}}{2x} - \frac{3{\mathrm{e}}^{-{x}^{2}}}{{2}^{2}{x}^{3}} + {\int }_{x}^{+\infty }{u}^{-4}{\mat... | Yes |
Problem 4 Verify this and obtain a formula for \( \cos z \) and \( \sin z \) . | ## A.2.4 Multiplication of Series and the Basic Property of the Exponential Function\n\nThe expression \( {\mathrm{e}}^{z} = {\mathrm{e}}^{x}\left( {\cos y + i\sin y}\right) \) for \( {\mathrm{e}}^{x + {iy}} \) can be naturally obtained from the relation \( {\mathrm{e}}^{x + {iy}} = {\mathrm{e}}^{x}{\mathrm{e}}^{iy} \)... | No |
Problem 1 Write down the metric \( {g}_{ij}\left( t\right) \mathrm{d}{t}^{i}\mathrm{\;d}{t}^{j} \) in each of these coordinate systems and find an orthonormal basis \( \left( {{e}_{1},{e}_{2},{e}_{3}}\right) \) . | Answer In Cartesian coordinates \( \left( {x, y, z}\right) \), cylindrical coordinates \( \left( {r,\varphi, z}\right) \), and spherical coordinates \( \left( {R,\varphi ,\theta }\right) \), the quadratic form \( {g}_{ij}\left( t\right) \mathrm{d}{t}^{i}\mathrm{\;d}{t}^{j} \) has the following form:\n\n\[ \mathrm{d}{s}... | Yes |
Problem 2 Write down in Cartesian, cylindrical, and spherical coordinates of the Euclidean space \( {\mathbb{R}}^{3} \) the explicit form of the 1 -form \( {\omega }_{A}^{1} = \langle A, \cdot \rangle \), corresponding to the vector \( A = {A}^{i}{e}_{i} \) . | Answer The 1-form \( {\omega }_{A}^{1} \) has the following form, in Cartesian coordinates \( \left( {x, y, z}\right) \) , cylindrical coordinates \( \left( {r,\varphi, z}\right) \), and spherical coordinates \( \left( {R,\varphi ,\theta }\right) \) of the Euclidean space \( {\mathbb{R}}^{3} \), respectively:\n\n\[ \n{... | Yes |
Problem 3 Write down the vector grad \( U = {A}_{e}^{i}{e}_{i} \) in Cartesian, cylindrical, and spherical coordinates of the Euclidean space \( {\mathbb{R}}^{3} \) . | Answer The vector grad \( U \) has the following form in Cartesian \( \left( {x, y, z}\right) \), cylindrical \( \left( {r,\theta, z}\right) \), and spherical \( \left( {R,\varphi ,\theta }\right) \) coordinates of the Euclidean space \( {\mathbb{R}}^{3} \), respectively:\n\n\[ \operatorname{grad}U = \frac{\partial U}{... | Yes |
Problem 4 Specify the explicit form of the 2 -form \( {\omega }_{B}^{2} = {\Omega }_{g}^{3}\left( {B,\ldots }\right) \) corresponding to the vector field \( B = {B}_{e}^{i}{e}_{i} \) in Cartesian, cylindrical, and spherical coordinates of the Euclidean space \( {\mathbb{R}}^{3} \) . | Answer The form \( {\omega }_{B}^{2} \) has the following form in Cartesian \( \left( {x, y, z}\right) \), cylindrical \( \left( {r,\theta, z}\right) \), and spherical \( \left( {R,\varphi ,\theta }\right) \) coordinates of the Euclidean space \( {\mathbb{R}}^{3} \) :\n\n\[ \n{\omega }_{B}^{2} = {B}_{x}\mathrm{\;d}y \l... | Yes |
Problem 5 Write down formulas to calculate the divergence of a vector field \( B = {B}_{e}^{i}{e}_{i} \) in Cartesian, cylindrical, and spherical coordinates of the Euclidean space \( {\mathbb{R}}^{3} \) . | Answer In Cartesian coordinates \( \left( {x, y, z}\right) \), cylindrical coordinates \( \left( {r,\varphi, z}\right) \), and spherical coordinates \( \left( {R,\varphi ,\theta }\right) \) of the Euclidean space \( {\mathbb{R}}^{3} \), the divergence div \( B \) of\nthe vector field \( B = {B}_{e}^{i}{e}_{i} \) can be... | Yes |
Problem 6 Write down the formula to calculate the curl of a vector field \( A = \) \( {A}_{e}^{1}{e}_{1} + {A}_{e}^{2}{e}_{2} + {A}_{3}^{3}{e}_{3} \) in Cartesian, cylindrical, and spherical coordinates of the Euclidean space \( {\mathbb{R}}^{3} \) . | Answer In Cartesian \( \left( {x, y, z}\right) \), cylindrical \( \left( {r,\varphi, z}\right) \), and spherical \( \left( {R,\varphi ,\theta }\right) \) coordinates of the Euclidean space, the curl (curl \( A \) ) of the vector field \( A = {A}_{e}^{1}{e}_{1} + {A}_{e}^{2}{e}_{2} + \n\n\( {A}_{3}^{3}{e}_{3} \) is calc... | Yes |
Theorem 1.1 (Helly, Hahn-Banach analytic form). Let \( p : E \rightarrow \mathbb{R} \) be a function satisfying \( {}^{1} \n\n(1)\n\n\[ \np\left( {\lambda x}\right) = {\lambda p}\left( x\right) \;\forall x \in E\;\text{ and }\;\forall \lambda > 0, \]\n\n(2)\n\n\[ \np\left( {x + y}\right) \leq p\left( x\right) + p\left(... | The proof of Theorem 1.1 depends on Zorn's lemma, which is a celebrated and very useful property of ordered sets. Before stating Zorn's lemma we must clarify some notions. Let \( P \) be a set with a (partial) order relation \( \leq \) . We say that a subset \( Q \subset P \) is totally ordered if for any pair \( \left... | Yes |
Proposition 1.5. The hyperplane \( H = \left\lbrack {f = \alpha }\right\rbrack \) is closed if and only if \( f \) is continuous. | Proof. It is clear that if \( f \) is continuous then \( H \) is closed. Conversely, let us assume that \( H \) is closed. The complement \( {H}^{c} \) of \( H \) is open and nonempty (since \( f \) does not vanish identically). Let \( {x}_{0} \in {H}^{c} \), so that \( f\left( {x}_{0}\right) \neq \alpha \), for exampl... | Yes |
Lemma 1.2. Let \( C \subset E \) be an open convex set with \( 0 \in C \) . For every \( x \in E \) set\n\n\[ p\left( x\right) = \inf \left\{ {\alpha > 0;{\alpha }^{-1}x \in C}\right\} \]\n\n( \( p \) is called the gauge of \( C \) or the Minkowski functional of \( C \) ).\n\nThen \( p \) satisfies (1),(2), and the fol... | Proof of Lemma 1.2. It is obvious that (1) holds.\n\nProof of (9). Let \( r > 0 \) be such that \( B\left( {0, r}\right) \subset C \) ; we clearly have\n\n\[ p\left( x\right) \leq \frac{1}{r}\parallel x\parallel \;\forall x \in E. \]\n\nProof of (10). First, suppose that \( x \in C \) ; since \( C \) is open, it follow... | Yes |
Lemma 1.3. Let \( C \subset E \) be a nonempty open convex set and let \( {x}_{0} \in E \) with \( {x}_{0} \notin C \) . Then there exists \( f \in {E}^{ \star } \) such that \( f\left( x\right) < f\left( {x}_{0}\right) \;\forall x \in C \) . In particular, the hyperplane \( \left\lbrack {f = f\left( {x}_{0}\right) }\r... | Proof of Lemma 1.3. After a translation we may always assume that \( 0 \in C \) . We may thus introduce the gauge \( p \) of \( C \) (see Lemma 1.2). Consider the linear subspace \( G = \mathbb{R}{x}_{0} \) and the linear functional \( g : G \rightarrow \mathbb{R} \) defined by\n\n\[ g\left( {t{x}_{0}}\right) = t,\;t \... | Yes |
Proposition 1.9. Let \( M \subset E \) be a linear subspace. Then\n\n\[ \left( {{M}^{ \bot }{)}^{ \bot } = \bar{M}}\right) \text{.} \]\n\nLet \( N \subset {E}^{ \star } \) be a linear subspace. Then\n\n\[ {\left( {N}^{ \bot }\right) }^{ \bot } \supset \bar{N} \] | Proof. It is clear that \( M \subset {\left( {M}^{ \bot }\right) }^{ \bot } \), and since \( {\left( {M}^{ \bot }\right) }^{ \bot } \) is closed we have \( \bar{M} \subset \) \( {\left( {M}^{ \bot }\right) }^{ \bot } \) . Conversely, let us show that \( {\left( {M}^{ \bot }\right) }^{ \bot } \subset \bar{M} \) . Suppos... | Yes |
Proposition 1.10. Assume that \( \varphi : E \rightarrow ( - \infty , + \infty \rbrack \) is convex l.s.c. and \( \varphi ≢ + \infty \) . Then \( {\varphi }^{ \star } ≢ + \infty \), and in particular, \( \varphi \) is bounded below by an affine continuous function. | Proof. Let \( {x}_{0} \in D\left( \varphi \right) \) and let \( {\lambda }_{0} < \varphi \left( {x}_{0}\right) \) . We apply Theorem 1.7 (Hahn-Banach, second geometric form) in the space \( E \times \mathbb{R} \) with \( A = \operatorname{epi}\varphi \) and \( B = \left\{ \left\lbrack {{x}_{0},{\lambda }_{0}}\right\rbr... | Yes |
Consider \( \varphi \left( x\right) = \parallel x\parallel \) . It is easy to check that | \[ {\varphi }^{ \star }\left( f\right) = \left\{ \begin{array}{ll} 0 & \text{ if }\parallel f\parallel \leq 1 \\ + \infty & \text{ if }\parallel f\parallel > 1 \end{array}\right. \] It follows that \[ {\varphi }^{\star \star }\left( x\right) = \mathop{\sup }\limits_{\substack{{f \in {E}^{ \star }} \\ {\parallel f\paral... | No |
Lemma 1.4. Let \( C \subset E \) be a convex set, then \( \operatorname{Int}C \) is convex. \( {}^{7} \) If, in addition, Int \( C \neq \varnothing \), then | For the proof of Lemma 1.4, see, e.g., Exercise 1.7. | No |
Let \( K \) be a nonempty convex set. We claim that for every \( {x}_{0} \in E \) we have\n\n(19)\n\n\[ \operatorname{dist}\left( {{x}_{0}, K}\right) = \mathop{\inf }\limits_{{x \in K}}\begin{Vmatrix}{x - {x}_{0}}\end{Vmatrix} = \mathop{\max }\limits_{\substack{{f \in {E}^{ \star }} \\ {\parallel f\parallel \leq 1} }}\... | Indeed, we have\n\n\[ \mathop{\inf }\limits_{{x \in K}}\begin{Vmatrix}{x - {x}_{0}}\end{Vmatrix} = \mathop{\inf }\limits_{{x \in E}}\{ \varphi \left( x\right) + \psi \left( x\right) \} \]\n\nwith \( \varphi \left( x\right) = \begin{Vmatrix}{x - {x}_{0}}\end{Vmatrix} \) and \( \psi \left( x\right) = {I}_{K}\left( x\righ... | Yes |
Let \( \varphi : E \rightarrow \mathbb{R} \) be convex and continuous and let \( M \subset E \) be a linear subspace. Then we have\n\n\[ \mathop{\inf }\limits_{{x \in M}}\varphi \left( x\right) = - \mathop{\min }\limits_{{f \in {M}^{ \bot }}}{\varphi }^{ \star }\left( f\right) \] | It suffices to apply Theorem 1.12 with \( \psi = {I}_{M} \) . | No |
Corollary 2.3. Let \( E \) and \( F \) be two Banach spaces. Let \( \left( {T}_{n}\right) \) be a sequence of continuous linear operators from \( E \) into \( F \) such that for every \( x \in E,{T}_{n}x \) converges (as \( n \rightarrow \infty \) ) to a limit denoted by \( {Tx} \) . Then we have 收敬性保i正了逐点有上界\n\n(a) \(... | Proof. (a) follows directly from Theorem 2.2, and thus there exists a constant \( c \) such that\n\n\[ \begin{Vmatrix}{{T}_{n}x}\end{Vmatrix} \leq c\parallel x\parallel \;\forall n,\;\forall x \in E. \]\n\nAt the limit we find\n\n\[ \parallel {Tx}\parallel \leq c\parallel x\parallel \;\forall x \in E. \]\n\nSince \( T ... | Yes |
Corollary 2.5. Let \( G \) be a Banach space and let \( {B}^{ \star } \) be a subset of \( {G}^{ \star } \) . Assume that\n\n) for every \( x \in G \) the set \( \left\langle {{B}^{ \star }, x}\right\rangle = \left\{ {\langle f, x\rangle ;f \in {B}^{ \star }}\right\} \) is bounded (in \( \mathbb{R} \) ).\n\nThen\n\n(6)... | Proof. Use Theorem 2.2 with \( E = G, F = \mathbb{R} \), and \( I = {B}^{ \star } \) . For every \( b \in {B}^{ \star } \) set\n\n\[ {T}_{b}\left( x\right) = \langle b, x\rangle \;\left( {x \in G = E}\right) . \]\n\nWe find that there exists a constant \( c \) such that\n\n\[ \left| {\langle b, x\rangle }\right| \leq c... | Yes |
Corollary 2.8. Let \( E \) be a vector space provided with two norms, \( {\begin{Vmatrix}{\parallel }_{1}\text{ and }\parallel \end{Vmatrix}}_{2} \) . Assume that \( E \) is a Banach space for both norms and that there exists a constant\n\n\[ \parallel x{\parallel }_{2} \leq C\parallel x{\parallel }_{1}\;\forall x \in ... | Proof of Corollary 2.8. Apply Corollary 2.7 with\n\n\[ E = \left( {E,\parallel {\parallel }_{1}}\right), F = \left( {E,\parallel {\parallel }_{2}}\right) \text{, and}T = I\text{.} \] | Yes |
Proposition 2.14. Let \( G \) and \( L \) be two closed subspaces in \( E \). Then\n\n\[ G \cap L = {\left( {G}^{ \bot } + {L}^{ \bot }\right) }^{ \bot } \] | Proof of (16). It is clear that \( G \cap L \subset {\left( {G}^{ \bot } + {L}^{ \bot }\right) }^{ \bot } \) ; indeed, if \( x \in G \cap L \) and \( f \in {G}^{ \bot } + {L}^{ \bot } \) then \( \langle f, x\rangle = 0 \) . Conversely, we have \( {G}^{ \bot } \subset {G}^{ \bot } + {L}^{ \bot } \) and thus \( {\left( {... | Yes |
Corollary 2.15. Let \( G \) and \( L \) be two closed subspaces in \( E \) . Then\n\n\[ \begin{array}{l} {\left( G \cap L\right) }^{ \bot } \supset \overline{{G}^{ \bot } + {L}^{ \bot }}\text{,由于 }{G}_{1}^{ \bot } + {L}^{ \bot } \subset {E}^{ * } \\ {\left( {G}^{ \bot } \cap {L}^{ \bot }\right) }^{ \bot } = \overline{G... | Proof. Use Propositions 1.9 and 2.14. | No |
Proposition 2.17. Let \( A : D\\left( A\\right) \\subset E \\rightarrow F \) be a densely defined unbounded linear operator. Then \( {A}^{ \\star } \) is closed, i.e., \( G\\left( {A}^{ \\star }\\right) \) is closed in \( {F}^{ \\star } \\times {E}^{ \\star } \) . | Proof. Let \( {v}_{n} \\in D\\left( {A}^{ \\star }\\right) \) be such that \( {v}_{n} \\rightarrow v \) in \( {F}^{ \\star } \) and \( {A}^{ \\star }{v}_{n} \\rightarrow f \) in \( {E}^{ \\star } \) . One has to check that (a) \( v \\in D\\left( {A}^{ \\star }\\right) \) and (b) \( {A}^{ \\star }v = f \) .\n\nWe have\n... | Yes |
Corollary 2.18. Let \( A : D\left( A\right) \subset E \rightarrow F \) be an unbounded linear operator that is densely defined and closed. Then\n\n(i) 容易验证N(A) 闭 \( N\left( A\right) = R{\left( {A}^{ \star }\right) }^{ \bot } \),\n\n(ii)\n\n\[ N\left( {A}^{ \star }\right) = R{\left( A\right) }^{ \bot },\]\n\n(iii)\n\n\[... | Proof. Note that (iii) and (iv) follow directly from (i) and (ii) combined with Proposition 1.9. There is a simple and direct proof of (i) and (ii) (see Exercise 2.18). However, it is instructive to relate these facts to Proposition 2.14 by the following device. Consider the space \( X = E \times F \), so that \( {X}^{... | No |
Problem 1. Construct a topology on \( X \) that makes all the maps \( {\left( {\varphi }_{i}\right) }_{i \in I} \) continuous. If possible, find a topology \( \mathcal{T} \) that is the most economical in the sense that it has the fewest open sets. | Note that if we equip \( X \) with the discrete topology (i.e., every subset of \( X \) is open), then every map \( {\varphi }_{i} \) is continuous; of course, this topology is far from being the | No |
Given a set \( X \) and a family \( {\left( {U}_{\lambda }\right) }_{\lambda \in \Lambda } \) of subsets in \( X \), construct the cheapest topology \( \mathcal{T} \) on \( X \) in which \( {U}_{\lambda } \) is open for all \( \lambda \in \Lambda \) . | In other words, we must find the cheapest family \( \mathcal{F} \) of subsets of \( X \) that is stable \( {}^{1} \) by \( { \cap }_{\text{finite }} \) and \( { \cup }_{\text{arbitrary }} \) and with the property that \( {U}_{\lambda } \in \mathcal{F} \) for every \( \lambda \in \Lambda \) . The construction goes as fo... | No |
Lemma 3.1. The family \( \mathcal{F} \) is stable under \( { \cap }_{\text{finite }} \) . | The proof of Lemma 3.1-a delightful exercise in set theory-is left to the reader; see e.g., G. Folland [2]. | No |
The unit sphere \( S = \{ x \in E;\parallel x\parallel = 1\} \), with \( E \) infinite-dimensional, is never closed in the weak topology \( \sigma \left( {E,{E}^{ \star }}\right) \) . More precisely, we have\n\n\[ \n{S}^{\sigma \left( {E,{E}^{ \star }}\right) } = {B}_{E}\text{,中}S\text{. 这 S在 OI E ,}{E}^{ \star }\text{... | First let us check that every \( {x}_{0} \in E \) with \( \begin{Vmatrix}{x}_{0}\end{Vmatrix} < 1 \) belongs to \( {\bar{S}}^{\sigma \left( {E,{E}^{ \star }}\right) } \) . Indeed, \( \bar{\top } < X \) . 在\n\n重要 let \( V \) be a neighborhood of \( {x}_{0} \) in \( \sigma \left( {E,{E}^{ \star }}\right) \) . We have to ... | Yes |
The unit ball \( U = \{ x \in E;\parallel x\parallel < 1\} \), with \( E \) infinite-dimensional, is never open in the weak topology \( \sigma \left( {E,{E}^{ \star }}\right) \) . | Suppose, by contradiction, that \( U \) is weakly open. Then its complement \( {U}^{c} = \{ x \in E;\parallel x\parallel \geq 1\} \) is weakly closed. It follows that \( S = {B}_{E} \cap {U}^{c} \) is also weakly closed; this contradicts Example 1. | Yes |
Corollary 3.8 (Mazur). Assume \( \left( {x}_{n}\right) \) converges weakly to \( x \) . Then there exists a sequence \( \left( {y}_{n}\right) \) made up of convex combinations of the \( {x}_{n} \) ’s that converges strongly to \( x \) . | Proof. Let \( C = \operatorname{conv}\left( {{ \cup }_{p = 1}^{\infty }\left\{ {x}_{p}\right\} }\right) \) denote the convex hull of the \( {x}_{n} \) ’s. Since \( x \) belongs to the weak closure of \( { \cup }_{p = 1}^{\infty }\left\{ {x}_{p}\right\} \) it belongs a fortiori to the weak closure of \( C \) . By Theore... | Yes |
Theorem 3.10. Let \( E \) and \( F \) be two Banach spaces and let \( T \) be a linear operator from \( E \) into \( F \) . Assume that \( T \) is continuous in the strong topologies. Then \( T \) is continuous from \( E \) weak \( \sigma \left( {E,{E}^{ \star }}\right) \) into \( F \) weak \( \sigma \left( {F,{F}^{ \s... | Proof. In view of Proposition 3.2 it suffices to check that for every \( f \in {F}^{ \star } \) the map\n\n存疑！ \( x \mapsto \langle f,{Tx}\rangle \) is continuous from \( E \) weak \( \sigma \left( {E,{E}^{ \star }}\right) \) into \( \mathbb{R} \) . But the map \( x \mapsto \langle f,{Tx}\rangle \) is a continuous line... | Yes |
Proposition 3.11. The weak* topology is Hausdorff. \( \Rightarrow \) 极限唯一 | Proof. Given \( {f}_{1},{f}_{2} \in {E}^{ \star } \) with \( {f}_{1} \neq {f}_{2} \) there exists some \( x \in E \) such that \( \left\langle {{f}_{1}, x}\right\rangle \neq \) \( \left\langle {{f}_{2}, x}\right\rangle \) (this does not use Hahn-Banach, but just the fact that \( {f}_{1} \neq {f}_{2} \) ). Assume for ex... | Yes |
Proposition 3.12. Let \( {f}_{0} \in {E}^{ \star } \) ; given a finite set \( \left\{ {{x}_{1},{x}_{2},\ldots ,{x}_{k}}\right\} \) in \( E \) and \( \varepsilon > 0 \) , consider\n\n\[ \nV = V\left( {{x}_{1},{x}_{2},\ldots ,{x}_{k};\varepsilon }\right) = \left\{ {f \in {E}^{ \star };\left| \left\langle {f - {f}_{0},{x}... | Proof. Same as the proof of Proposition 3.4. | No |
Lemma 3.2. Let \( X \) be a vector space and let \( \varphi ,{\varphi }_{1},{\varphi }_{2},\ldots ,{\varphi }_{k} \) be \( \left( {k + 1}\right) \) linear functionals on \( X \) such that \[ \left\lbrack {{\varphi }_{i}\left( v\right) = 0\;\forall i = 1,2,\ldots, k}\right\rbrack \Rightarrow \left\lbrack {\varphi \left(... | Proof of Lemma 3.2. Consider the map \( F : X \rightarrow {\mathbb{R}}^{k + 1} \) defined by \[ F\left( u\right) = \left\lbrack {\varphi \left( u\right) ,{\varphi }_{1}\left( u\right) ,{\varphi }_{2}\left( u\right) ,\ldots ,{\varphi }_{k}\left( u\right) }\right\rbrack . \] It follows from assumption (2) that \( a = \le... | Yes |
Lemma 3.3 (Helly). Let \( E \) be a Banach space. Let \( {f}_{1},{f}_{2},\ldots ,{f}_{k} \) be given in \( {E}^{ \star } \) and let \( {\gamma }_{1},{\gamma }_{2},\ldots ,{\gamma }_{k} \) be given in \( \mathbb{R} \) . The following properties are equivalent:\n\n(i) \( \forall \varepsilon > 0\exists {x}_{\varepsilon } ... | Proof. (i) \( \Rightarrow \) (ii). Fix \( {\beta }_{1},{\beta }_{2},\ldots ,{\beta }_{k} \) in \( \mathbb{R} \) and let \( S = \mathop{\sum }\limits_{{i = 1}}^{k}\left| {\beta }_{i}\right| \) . It follows from (i)\n\nthat\n\[ \left| {\mathop{\sum }\limits_{{i = 1}}^{k}{\beta }_{i}\left\langle {{f}_{i},{x}_{\varepsilon ... | Yes |
Lemma 3.4 (Goldstine). Let \( E \) be any Banach space. Then \( J\left( {B}_{E}\right) \) is dense in \( {B}_{{E}^{\star \star }} \) with respect to the topology \( \sigma \left( {{E}^{\star \star },{E}^{ \star }}\right) \), and consequently \( J\left( E\right) \) is dense in \( {E}^{\star \star } \) in the topology \(... | Proof. Let \( \xi \in {B}_{{E}^{\star \star }} \) and let \( V \) be a neighborhood of \( \xi \) for the topology \( \sigma \left( {{E}^{\star \star },{E}^{ \star }}\right) \) . We must prove that \( V \cap J\left( {B}_{E}\right) \neq \varnothing \) . As usual, we may assume that \( V \) is of the form \[ V = \left\{ {... | Yes |
Corollary 3.21. A Banach space \( E \) is reflexive (if and only if) its dual space \( {E}^{ \star } \) is reflexive. | Proof. E reflexive \( \Rightarrow {E}^{ \star } \) reflexive. The idea of the proof is simple, since, roughly speaking, we have that \( {E}^{\star \star } = E \Rightarrow {E}^{\star \star \star } = {E}^{ \star } \) . More precisely, let \( J \) be the canonical isomorphism from \( E \) into \( {E}^{\star \star } \) . L... | Yes |
Theorem 3.24. Let \( E \) and \( F \) be two reflexive Banach spaces. Let \( A : D\left( A\right) \subset E \rightarrow \) 伴随算子 \( F \) be an unbounded linear operator that is densely defined and closed. Then \( \underline{D\left( {A}^{ \star }\right) } \) 一直是闭算子 is dense in \( {F}^{ \star } \) . Thus \( {A}^{\star \st... | Proof. \[ {\varphi }_{D\left( {A}^{x}\right) } = 0 \] 1. \( D\left( {A}^{ \star }\right) \) is dense in \( {F}^{ \star } \) . Let \( \varphi \) be a continuous linear functional on \( {F}^{ \star } \) that vanishes on \( D\left( {A}^{ \star }\right) \) . In view of Corollary 1.8 it suffices to prove that \( \varphi \eq... | No |
Proposition 3.25. Let \( E \) be a separable metric space and let \( F \subset E \) be any subset. Then \( F \) is also separable. | Proof. Let \( \left( {u}_{n}\right) \) be a countable dense subset of \( E \) . Let \( \left( {r}_{m}\right) \) be any sequence of positive numbers such that \( {r}_{m} \rightarrow 0 \) . Choose any point \( {a}_{m, n} \in B\left( {{u}_{n},{r}_{m}}\right) \cap F \) whenever this set is nonempty. The set \( \left( {a}_{... | Yes |
Theorem 3.26. Let \( E \) be a Banach space such that \( {E}^{ \star } \) is separable. Then \( E \) is separable. | Proof. Let \( {\left( {f}_{n}\right) }_{n \geq 1} \) be countable and dense in \( {E}^{ \star } \) . Since\n\n\[ \begin{Vmatrix}{f}_{n}\end{Vmatrix} = \mathop{\sup }\limits_{\substack{{x \in E} \\ {\parallel x\parallel \leq 1} }}\left\langle {{f}_{n}, x}\right\rangle \]\n\nwe can find some \( {x}_{n} \in E \) such that... | Yes |
Corollary 3.27. Let \( E \) be a Banach space. Then\n\n\[ \left\lbrack {E\text{reflexive and separable}}\right\rbrack \Leftrightarrow \left\lbrack {{E}^{ \star }\text{reflexive and separable}}\right\rbrack \text{.} \] | Proof. We already know (Corollary 3.21 and Theorem 3.26) that\n\n\[ \left\lbrack {{E}^{ \star }\text{reflexive and separable}}\right\rbrack \Rightarrow \left\lbrack {E\text{reflexive and separable}}\right\rbrack \text{.} \]\n\nConversely, if \( E \) is reflexive and separable, so is \( {E}^{\star \star } = J\left( E\ri... | Yes |
Let \( E = {\mathbb{R}}^{2} \) . The norm \( \parallel x{\parallel }_{2} = {\left\lbrack {\left| {x}_{1}\right| }^{2} + {\left| {x}_{2}\right| }^{2}\right\rbrack }^{1/2} \) is uniformly convex, while the norm \( \parallel x{\parallel }_{1} = \left| {x}_{1}\right| + \left| {x}_{2}\right| \) and the norm \( \parallel x{\... | This can be easily seen by staring at the unit balls, as shown in Figure 3. | No |
Theorem 3.31 (Milman-Pettis). Every uniformly convex Banach space is reflexive. | Proof. Let \( \xi \in {E}^{\star \star } \) with \( \parallel \xi \parallel = 1 \) . We have to show that \( \xi \in J\left( {B}_{E}\right) \) . Since \( J\left( {B}_{E}\right) \) is closed in \( {E}^{\star \star } \) in the strong topology, it suffices to prove that\n\n(7)\n\nFix \( \varepsilon > 0 \) and let \( \delt... | Yes |
Proposition 3.32. Assume that \( E \) is a uniformly convex Banach space. Let \( \left( {x}_{n}\right) \) be a sequence in \( E \) such that \( {x}_{n} \rightharpoonup x \) weakly \( \sigma \left( {E,{E}^{ \star }}\right) \) and \( \begin{Vmatrix}{x}_{n}\end{Vmatrix} \rightarrow \parallel x\parallel \). Then \( {x}_{n}... | Proof. We may always assume that \( x \neq 0 \) (otherwise the conclusion is obvious). Set\n\n\[ \n{\lambda }_{n} = \max \left( {\begin{Vmatrix}{x}_{n}\end{Vmatrix},\parallel x\parallel }\right) ,{y}_{n} = {\lambda }_{n}^{-1}{x}_{n}\text{, and}y = \parallel x{\parallel }^{-1}x\text{,} \n\]\n\nso that \( {\lambda }_{n} ... | Yes |
Theorem 4.3 (density). The space \( {C}_{c}\left( {\mathbb{R}}^{N}\right) \) is dense in \( {L}^{1}\left( {\mathbb{R}}^{N}\right) \) ; i.e., | \[ \forall f \in {L}^{1}\left( {\mathbb{R}}^{N}\right) \;\forall \varepsilon > 0\;\exists {f}_{1} \in {C}_{c}\left( {\mathbb{R}}^{N}\right) \text{ such that }{\begin{Vmatrix}f - {f}_{1}\end{Vmatrix}}_{1} \leq \varepsilon . \] | Yes |
Theorem 4.7. \( {L}^{p} \) is a vector space and \( \parallel {\parallel }_{p} \) is a norm for any \( p,1 \leq p \leq \infty \) . | Proof. The cases \( p = 1 \) and \( p = \infty \) are clear. Therefore we assume \( 1 < p < \infty \) and let \( f, g \in {L}^{p} \) . We have\n\n\[ \n{\left| f\left( x\right) + g\left( x\right) \right| }^{p} \leq {\left( \left| f\left( x\right) \right| + \left| g\left( x\right) \right| \right) }^{p} \leq {2}^{p}\left(... | Yes |
Theorem 4.9. Let \( \left( {f}_{n}\right) \) be a sequence in \( {L}^{p} \) and let \( f \in {L}^{p} \) be such that \( {\begin{Vmatrix}{f}_{n} - f\end{Vmatrix}}_{p} \) \( \rightarrow 0 \) .\n\nThen, there exist a subsequence \( \left( {f}_{{n}_{k}}\right) \) and a function \( h \in {L}^{p} \) such that\n\n(a) \( {f}_{... | Proof. The conclusion is obvious when \( p = \infty \) . Thus we assume \( 1 \leq p < \infty \) . Since \( \left( {f}_{n}\right) \) is a Cauchy sequence we may go back to the proof of Theorem 4.8 and consider a subsequence \( \left( {f}_{{n}_{k}}\right) \) -denoted by \( \left( {f}_{k}\right) \) -satisfying (6), such t... | Yes |
Theorem 4.12. The space \( {C}_{c}\left( {\mathbb{R}}^{N}\right) \) is dense in \( {L}^{p}\left( {\mathbb{R}}^{N}\right) \) for any \( p,1 \leq p < \infty \) . | Proof. First, we claim that given \( f \in {L}^{p}\left( {\mathbb{R}}^{N}\right) \) and \( \varepsilon > 0 \) there exist a function \( g \in {L}^{\infty }\left( {\mathbb{R}}^{N}\right) \) and a compact set \( K \) in \( {\mathbb{R}}^{N} \) such that \( g = 0 \) outside \( K \) and\n\n(11)\n\n\[ \parallel f - g{\parall... | Yes |
Theorem 4.13. Assume that \( \Omega \) is a separable measure space. Then \( {L}^{p}\left( \Omega \right) \) is separable for any \( p,1 \leq p < \infty \) . | Proof of Theorem 4.13 when \( \Omega = {\mathbb{R}}^{N} \) . Let \( \mathcal{R} \) denote the countable family of sets in \( {\mathbb{R}}^{N} \) of the form \( R = \mathop{\prod }\limits_{{k = 1}}^{N}\left( {{a}_{k},{b}_{k}}\right) \) with \( {a}_{k},{b}_{k} \in \mathbb{Q} \) . Let \( \mathcal{E} \) denote the vector s... | No |
Lemma 4.2. Let \( E \) be a Banach space. Assume that there exists a family \( {\left( {O}_{i}\right) }_{i \in I} \) such that\n\n(i) for each \( i \in I,{O}_{i} \) is a nonempty open subset of \( E \) ,\n\n(ii) \( {O}_{i} \cap {O}_{j} = \varnothing \) if \( i \neq j \) ,\n\n(iii) I is uncountable.\n\nThen \( E \) is n... | Proof of Lemma 4.2. Suppose, by contradiction, that \( E \) is separable. Let \( {\left( {u}_{n}\right) }_{n \in \mathbb{N}} \) denote a dense countable set in \( E \) . For each \( i \in I \), the set \( {O}_{i} \cap {\left( {u}_{n}\right) }_{n \in \mathbb{N}} \neq \varnothing \) and we may choose \( n\left( i\right) ... | Yes |
Proposition 4.16. Let \( f \in {L}^{1}\left( {\mathbb{R}}^{N}\right), g \in {L}^{p}\left( {\mathbb{R}}^{N}\right) \) and \( h \in {L}^{{p}^{\prime }}\left( {\mathbb{R}}^{N}\right) \) . Then we have\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}\left( {f \star g}\right) h = {\int }_{{\mathbb{R}}^{N}}g\left( {\breve{f} \star h}\righ... | Proof. The function \( F\left( {x, y}\right) = f\left( {x - y}\right) g\left( y\right) h\left( x\right) \) belongs to \( {L}^{1}\left( {{\mathbb{R}}^{N} \times {\mathbb{R}}^{N}}\right) \) since\n\n\[ \n\int \left| {h\left( x\right) }\right| {dx}\int \left| {f\left( {x - y}\right) }\right| \left| {g\left( y\right) }\rig... | Yes |
Proposition 4.17 (and definition of the support). Let \( f : {\mathbb{R}}^{N} \rightarrow \mathbb{R} \) be any function. Consider the family \( {\left( {\omega }_{i}\right) }_{i \in I} \) of all open sets on \( {\mathbb{R}}^{N} \) such that for each \( i \in I, f = 0 \) a.e. on \( {\omega }_{i} \) . Set \( \omega = \ma... | Proof of Proposition 4.17. Since the set \( I \) need not be countable it is not clear that \( f = 0 \) a.e. on \( \omega \) . However we may recover the countable case as follows. There is a countable family \( \left( {O}_{n}\right) \) of open sets in \( {\mathbb{R}}^{N} \) such that every open set on \( {\mathbb{R}}^... | Yes |
Proposition 4.19. Let \( f \in {C}_{c}\left( {\mathbb{R}}^{N}\right) \) and \( g \in {L}_{\mathrm{{loc}}}^{1}\left( {\mathbb{R}}^{N}\right) \) . Then \( \left( {f \star g}\right) \left( x\right) \) is well defined for every \( x \in {\mathbb{R}}^{N} \), and, moreover, \( \left( {f \star g}\right) \in C\left( {\mathbb{R... | Proof. Note that for every \( x \in {\mathbb{R}}^{N} \) the function \( y \mapsto f\left( {x - y}\right) g\left( y\right) \) is integrable on \( {\mathbb{R}}^{N} \) and therefore \( \left( {f \star g}\right) \left( x\right) \) is defined for every \( x \in {\mathbb{R}}^{N} \) . Let \( {x}_{n} \rightarrow x \) and let \... | Yes |
Proposition 4.21. Assume \( f \in C\left( {\mathbb{R}}^{N}\right) \) . Then \( \left( {{\rho }_{n} \star f}\right) \underset{n \rightarrow \infty }{ \rightarrow }f \) uniformly on compact sets of \( {\mathbb{R}}^{N} \) . | Proof. \( {}^{4} \) Let \( K \subset {\mathbb{R}}^{N} \) be a fixed compact set. Given \( \varepsilon > 0 \) there exists \( \delta > 0 \) (depending on \( K \) and \( \varepsilon \) ) such that\n\n\[ \left| {f\left( {x - y}\right) - f\left( x\right) }\right| < \varepsilon \;\forall x \in K,\;\forall y \in B\left( {0,\... | Yes |
Corollary 4.24. Let \( \Omega \subset {\mathbb{R}}^{N} \) be an open set and let \( u \in {L}_{\text{loc }}^{1}\left( \Omega \right) \) be such that\n\n\[ \int {uf} = 0\;\forall f \in {C}_{c}^{\infty }\left( \Omega \right) \]\n\nThen \( u = 0 \) a.e. on \( \Omega \) . | Proof. Let \( g \in {L}^{\infty }\left( {\mathbb{R}}^{N}\right) \) be a function such that supp \( g \) is a compact set contained in \( \Omega \) . Set \( {g}_{n} = {\rho }_{n} \star g \), so that \( {g}_{n} \in {C}_{c}^{\infty }\left( \Omega \right) \) provided \( n \) is large enough. Therefore we have\n\n(19)\n\n\[... | Yes |
Corollary 4.27. Let \( \mathcal{F} \) be a bounded set in \( {L}^{p}\left( {\mathbb{R}}^{N}\right) \) with \( 1 \leq p < \infty \) . Assume (22) and also\n\n(27)\n\n\[ \left\{ \begin{array}{l} \forall \varepsilon > 0\exists \Omega \subset {\mathbb{R}}^{N},\text{ bounded, measurable such that } \\ \parallel f{\parallel ... | Proof. Given \( \varepsilon > 0 \) we fix \( \Omega \subset {\mathbb{R}}^{N} \) bounded measurable such that (27) holds. By Theorem 4.26 we know that \( {\mathcal{F}}_{\mid \Omega } \) has compact closure in \( {L}^{p}\left( \Omega \right) \) . Hence we may cover \( {\mathcal{F}}_{\mid \Omega } \) with a finite number ... | Yes |
Corollary 4.28. Let \( G \) be a fixed function in \( {L}^{1}\left( {\mathbb{R}}^{N}\right) \) and let\n\n\[ \mathcal{F} = G \star \mathcal{B} \]\n\nwhere \( \mathcal{B} \) is a bounded set in \( {L}^{p}\left( {\mathbb{R}}^{N}\right) \) with \( 1 \leq p < \infty \) . Then \( {\mathcal{F}}_{\mid \Omega } \) has compact ... | Proof. Clearly \( \mathcal{F} \) is bounded in \( {L}^{p}\left( {\mathbb{R}}^{N}\right) \) . On the other hand, if we write \( f = G \star u \) with \( u \in \mathcal{B} \) we have\n\n\[ {\begin{Vmatrix}{\tau }_{h}f - f\end{Vmatrix}}_{p} = {\begin{Vmatrix}\left( {\tau }_{h}G - G\right) \star u\end{Vmatrix}}_{p} \leq C{... | No |
Lemma 4.3. Let \( G \in {L}^{q}\left( {\mathbb{R}}^{N}\right) \) with \( 1 \leq q < \infty \) . Then \[ \mathop{\lim }\limits_{{h \rightarrow 0}}{\begin{Vmatrix}{\tau }_{h}G - G\end{Vmatrix}}_{q} = 0. \] | Proof. Given \( \varepsilon > 0 \), there exists (by Theorem 4.12) a function \( {G}_{1} \in {C}_{c}\left( {\mathbb{R}}^{N}\right) \) such that \( {\begin{Vmatrix}G - {G}_{1}\end{Vmatrix}}_{q} < \varepsilon \) . We write \[ {\begin{Vmatrix}{\tau }_{h}G - G\end{Vmatrix}}_{q} \leq {\begin{Vmatrix}{\tau }_{h}G - {\tau }_{... | Yes |
Theorem 4.33 (Young). Assume \( f \in {L}^{p}\left( {\mathbb{R}}^{N}\right) \) and \( g \in {L}^{q}\left( {\mathbb{R}}^{N}\right) \) with \( 1 \leq p \leq \infty \) , \( 1 \leq q \leq \infty \) and \( \frac{1}{r} = \frac{1}{p} + \frac{1}{q} - 1 \geq 0 \) . Then \( f \star g \in {L}^{r}\left( {\mathbb{R}}^{N}\right) \) ... | For a proof see, e.g., Exercise 4.30. | No |
Proposition 5.3. Let \( K \subset H \) be a nonempty closed convex set. Then \( {P}_{K} \) does not increase distance, i.e.，投影映射一致 Lipschtz 连续\n\n\[ \left| {{P}_{K}{f}_{1} - {P}_{K}{f}_{2}}\right| \leq \left| {{f}_{1} - {f}_{2}}\right| \;\forall {f}_{1},{f}_{2} \in H. \] | Proof. Set \( {u}_{1} = {P}_{K}{f}_{1} \) and \( {u}_{2} = {P}_{K}{f}_{2} \) . We have\n\n(6)\n\n\[ \left( {{f}_{1} - {u}_{1}, v - {u}_{1}}\right) \leq 0\;\forall v \in K \]\n\n(7)\n\[ \left( {{f}_{2} - {u}_{2}, v - {u}_{2}}\right) \leq 0\;\forall v \in K. \]\n\nChoosing \( v = {u}_{2} \) in (6) and \( v = {u}_{1} \) i... | Yes |
Assume that \( M \subset H \) is a closed linear subspace. Let \( f \in H \) . Then \( u = {P}_{M}f \) is characterized by\n\n\[ u \in M\;{and}\;\left( {f - u, v}\right) = 0\;\forall v \in M. \] | Proof. By (3) we have\n\n\[ \left( {f - u, v - u}\right) \leq 0\;\forall v \in M \]\n\nand thus\n\n\[ \left( {f - u,{tv} - u}\right) \leq 0\;\forall v \in M,\;\forall t \in \mathbb{R}. \]\n\nIt follows that (8) holds.\n\nConversely, if \( u \) satisfies (8) we have\n\n\[ \left( {f - u, v - u}\right) = 0\;\forall v \in ... | Yes |
Theorem 5.6 (Stampacchia). Assume that \( a\left( {u, v}\right) \) is a continuous coercive bilinear form on \( H \) . Let \( K \subset H \) be a nonempty closed and convex subset. Then, given any \( \varphi \in {H}^{ \star } \), there exists a unique element \( u \in K \) such that\n\n\[ a\left( {u, v - u}\right) \geq... | Proof of Theorem 5.6. From the Riesz-Fréchet representation theorem (Theorem 5.5) we know that there exists a unique \( f \in H \) such that\n\n\[ \underbrace{\langle \varphi, v\rangle = \left( {f, v}\right) }\;\forall v \in H. \]\n\nOn the other hand, if we fix \( u \in H \), the map \( v \mapsto a\left( {u, v}\right)... | Yes |
Lemma 5.1. Assume that \( \\left( {v}_{n}\\right) \) is any sequence in \( H \) such that\n\n(21)\n\n\[ \n\\begin{array}{l} \\left( {{v}_{m},{v}_{n}}\\right) = 0\\;\\forall m \\neq n, \\\\ \\mathop{\\sum }\\limits_{{k = 1}}^{\\infty }{\\left| {v}_{k}\\right| }^{2} < \\infty . \\end{array} \n\]\n\n(22)\n\nSet\n\n---\n\n... | Proof of Lemma 5.1. Note that for \( m > n \) we have\n\n\[ \n\\left( {{\\left| {S}_{m} - {S}_{n}\\right| }^{2} = \\mathop{\\sum }\\limits_{{k = n + 1}}^{m}{\\left| {v}_{k}\\right| }^{2}}\\right. \n\]\n\nIt follows that \( {S}_{n} \) is a Cauchy sequence and thus \( S = \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty... | Yes |
Theorem 6.1. The set \( \mathcal{K}\left( {E, F}\right) \) is a closed linear subspace of \( \mathcal{L}\left( {E, F}\right) \) (in the topology associated to the norm \( \parallel {\parallel }_{\mathcal{L}\left( {E, F}\right) } \) ). | Proof. Clearly the sum of two compact operators is a compact operator. Suppose that \( \left( {T}_{n}\right) \) is a sequence of compact operators and \( T \) is a bounded operator such that \( {\begin{Vmatrix}{T}_{n} - T\end{Vmatrix}}_{\mathcal{L}\left( {E, F}\right) } \rightarrow 0 \) . We claim that \( T \) is a com... | Yes |
Corollary 6.2. Let \( \left( {T}_{n}\right) \) be a sequence of finite-rank operators and let \( T \in \mathcal{L}\left( {E, F}\right) \) be such that \( {\begin{Vmatrix}{T}_{n} - T\end{Vmatrix}}_{\mathcal{L}\left( {E, F}\right) } \rightarrow 0 \) . Then \( T \in \mathcal{K}\left( {E, F}\right) \) . | proof: 首先有 \( T \in K\left( {E, F}\right) \subseteq L\left( {E, F}\right) \)\n\n\( \Rightarrow T{x}_{n} \rightharpoonup {Tx} \) 反设 \( T{x}_{n} \rightarrow {Tx} \)\n\n\( \Rightarrow \left| {T{x}_{{n}_{k}} - {Tx}}\right| > \varepsilon \vee {n}_{k} \) . 对某-3到 \( {n}_{k} \)\n\n而弱收敛 知 ‖ \( {\chi }_{n} \parallel \) -致有界 \( \... | No |
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