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Proposition 4.6. Let \( X \) be a connected \( {C}^{1} \) manifold. If \( X \) is not orientable, then there exists a covering \( {X}^{\prime } \rightarrow X \) of degree 2 such that \( {X}^{\prime } is orientable. | Sketch of Proof. Suppose first that \( X \) is simply connected. Let \( x \in X \) . Fix a chart \( \left( {{U}_{0},{\varphi }_{0}}\right) \) at \( x \) such that the image of the chart is an open ball in euclidean space. Let \( y \) be any point of \( X \), and let \( \alpha : \left\lbrack {a, b}\right\rbrack \rightar... | No |
Theorem 4.8. Let \( \pi : X \rightarrow Z \) be a submersion. Let \( \Omega \) be a volume form on \( X \) and \( \omega \) a volume form on \( Z \) . Let \( \Omega = \eta \otimes \omega \) . Let \( \widetilde{\eta } \) be a form on \( X \) , of the same degree as \( \eta \), restricting to \( \eta \) on the fibers. Th... | Proof. The proposition is local, since by using a partition of unity, we are reduced to the case when the support of \( f \) is in a given neighborhood of a point. Then the submersion is represented in a chart as a projection \( U \times W \rightarrow W \), where \( U, W \) are open in \( {\mathbf{R}}^{p} \) and \( {\m... | Yes |
Lemma 5.2. Let \( f \in {C}_{c}\left( G\right) \) . If \( {f}^{H} = 0 \), that is\n\n\[ \n{\int }_{H}f\left( {xh}\right) {dh} = 0 \n\]\n\nfor all \( x \in G \), then\n\n\[ \n{\int }_{G}f\left( x\right) {dx} = 0 \n\] | Proof. For all \( \varphi \in {C}_{c}\left( G\right) \), we have:\n\n\[ \n{\int }_{G}\varphi \left( x\right) \left( {{\int }_{H}f\left( {xh}\right) {dh}}\right) {dx} = {\int }_{G}\left( {{\int }_{H}\varphi \left( x\right) f\left( {xh}\right) {dh}}\right) {dx} \n\]\n\n\[ \n= {\int }_{H}\left( {{\int }_{G}\varphi \left( ... | Yes |
Corollary 2.3. Let \( X \) be a \( {C}^{2} \) oriented manifold, of dimension \( n \), and let \( \omega \) be an \( \left( {n - 1}\right) \) -form on \( X \), of class \( {C}^{1} \) . Assume that \( \omega \) has almost compact support, and that the measures associated with \( \left| {d\omega }\right| \) on \( X \) an... | Proof. By our standard form of Stokes' theorem we have\n\n\[ \n{\int }_{\partial X}{g}_{k}\omega = {\int }_{X}d\left( {{g}_{k}\omega }\right) = {\int }_{X}d{g}_{k} \land \omega + {\int }_{X}{g}_{k}{d\omega }. \n\]\n\nWe estimate the left-hand side by\n\n\[ \n\left| {{\int }_{\partial X}\omega - {\int }_{\partial X}{g}_... | Yes |
Theorem 3.1 (Stokes’ Theorem with Singularities). Let \( X \) be an oriented, \( {C}^{3} \) submanifold without boundary of \( {\mathbf{R}}^{N} \). Let \( \dim X = n \). Let \( \omega \) be an \( \left( {n - 1}\right) \) -form of class \( {C}^{1} \) on an open neighborhood of \( \bar{X} \) in \( {\mathbf{R}}^{N} \), an... | Proof. Let \( U,\left\{ {U}_{k}\right\} \), and \( \left\{ {g}_{k}\right\} \) satisfy conditions NEG 1 and NEG 2. Then \( {g}_{k}\omega \) is 0 on an open neighborhood of \( S \), and since \( \omega \) is assumed to have compact support, one verifies immediately that\n\n\[ \n\left( {\operatorname{supp}{g}_{k}\omega }\... | Yes |
Lemma 3.2. Let \( S \) be a compact subset of \( {\mathbf{R}}^{n} \). Let \( {U}_{k} \) be the open set of points \( x \) such that \( d\left( {x, S}\right) < 2/k \). There exists a \( {C}^{\infty } \) function \( {g}_{k} \) on \( {\mathbf{R}}^{N} \) which is equal to 0 in some open neighborhood of \( S \), equal to 1 ... | Proof. Let \( \varphi \) be a \( {C}^{\infty } \) function such that \( 0 \leqq \varphi \leqq 1 \), and\n\n\[ \varphi \left( x\right) = 0\;\text{ if }\;0 \leqq \parallel x\parallel \leqq \frac{1}{2}, \]\n\n\[ \varphi \left( x\right) = 1\;\text{ if }\;1 \leqq \parallel x\parallel . \]\n\nWe use \( \parallel \parallel \)... | Yes |
Theorem 3.3. Let \( X \) be an open subset of \( {\mathbf{R}}^{n} \). Let \( S \) be the set of singular points in the closure of \( X \), and assume that \( S \) is the finite union of \( {C}^{1} \) images of m-rectangles with \( m \leqq n - 2 \). Let \( \omega \) be an \( \left( {n - 1}\right) \)-form defined on an o... | Proof. Immediate from our two criteria and Theorem 3.2. | No |
Theorem 1.2. Let \( X \) be a manifold without boundary, of dimension \( n \). Suppose that \( X \) is orientable and connected. Then the map \[ \omega \mapsto {\int }_{X}\omega \] induces an isomorphism of \( {H}_{c}^{n}\left( X\right) \) with \( \mathbf{R} \) itself. | Proof. By Stokes' theorem (Chapter XVII, Corollary 2.2) the integral vanishes on exact forms (with compact support), and hence induces an R-linear map of \( {H}_{c}^{n}\left( X\right) \) into \( \mathbf{R} \). The theorem amounts to proving the converse statement: if \[ {\int }_{X}\omega = 0 \] then there exists some \... | Yes |
Lemma 1.4. Let \( \omega \) be an \( \left( {n - 1}\right) \) -form on \( {I}^{n - 1} \) whose coefficient is a function of \( n \) variables \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) so\n\n\[ \omega \left( x\right) = f\left( {{x}_{1},\ldots ,{x}_{n}}\right) d{x}_{1} \land \cdots \land d{x}_{n - 1}. \]\n\n(Of cours... | Proof. By induction. We first prove the theorem when \( n - 1 = 1 \) . First we carry out the proof leaving out the extra variable, just to see what's going on. So let\n\n\[ \omega \left( x\right) = f\left( x\right) {dx} \]\n\nwhere \( f \) has compact support in the open interval \( \left( {0,1}\right) \) . This means... | Yes |
Lemma 1.5. Let \( U \) be an open subset of \( X \), isomorphic to \( {I}^{n} \) . Let \( \psi \in {\mathcal{A}}_{c}^{n}\left( U\right) \) be such that\n\n\[{\int }_{U}\psi \neq 0\]\n\nLet \( \omega \in {\mathcal{A}}_{c}^{n}\left( U\right) \) . Then there exists \( c \in \mathbf{R} \) and \( \eta \in {\mathcal{A}}_{c}^... | Proof. We take \( c = {\int }_{U}\omega /{\int }_{U}\psi \) and apply Lemma 1.3 to \( \omega - {c\psi } \) . | Yes |
Lemma 1.6. Assume that \( X \) is connected and oriented. Let \( U,\psi \) be as in Lemma 1.5. Let \( V \) be the set of points \( x \in X \) having the following property. There exists a neighborhood \( U\left( x\right) \) of \( x \) isomorphic to \( {I}^{n} \) such that for every \( \omega \in {\mathcal{A}}_{c}^{n}\l... | Proof. Lemma 1.5 asserts that \( V \supset U \) . Since \( X \) is connected, it suffices to prove that \( V \) is both open and closed. It is immediate from the definition of \( V \) that \( V \) is open, so there remains to prove its closure. Let \( z \) be in the closure of \( V \) . Let \( W \) be a neighborhood of... | Yes |
Theorem 2.2. Let \( X \) be a compact, connected oriented manifold of dimension \( n \) . Let \( \omega ,\psi \in {\mathcal{A}}^{n}\left( X\right) \left( { = {\mathcal{A}}_{c}^{n}\left( X\right) }\right) \) be volume forms such that\n\n\[ \n{\int }_{X}\omega = {\int }_{X}\psi \n\]\n\nThen there exists an automorphism \... | Proof. Let\n\n\[ \n{\omega }_{s} = \left( {1 - s}\right) \omega + {s\psi }\;\text{ for }\;0 \leqq s \leqq 1. \n\]\n\nThen \( {\omega }_{s} \) is a volume form for each \( s \), and in particular is non-singular. By Theorem 1.1, there exists \( \eta \in {\mathcal{A}}^{n - 1}\left( X\right) \) such that \( \psi - \omega ... | Yes |
Theorem 3.1 (Divergence Theorem).\n\n\[ \n{\int }_{X}{\mathcal{L}}_{\xi }\Omega = {\int }_{\partial X}\Omega \circ \xi \n\] | Suppose that \( \left( {X, g}\right) \) is a Riemannian manifold, assumed oriented for simplicity. We let \( \Omega \) or \( {\operatorname{vol}}_{g} \) be the volume form defined in Chapter XV,\n\n§1. Let \( \omega \) be the canonical Riemannian volume form on \( \partial X \) for the metric induced by \( g \) on the ... | Yes |
Theorem 3.4 (Green’s Formula). Let \( \left( {X, g}\right) \) be an oriented Riemannian manifold possibly with boundary, and let \( \varphi ,\psi \) be functions on \( X \) with compact support. Let \( \omega \) be the canonical volume form associated with the induced metric on the boundary. Then\n\n\[ \n{\int }_{X}\le... | Proof. From formula (4) we get\n\n\[ \n{d}^{ * }\left( {\varphi d\psi }\right) = {\varphi \Delta \psi } - \langle {d\varphi },{d\psi }{\rangle }_{g},\n\]\n\nwhence\n\n\[ \n{\varphi \Delta \psi } - {\psi \Delta \varphi } = {d}^{ * }\left( {\varphi d\psi }\right) - {d}^{ * }\left( {\psi d\varphi }\right)\n\]\n\n\[ \n= - ... | Yes |
Corollary 3.5 (E. Hopf). Let \( X \) be a Riemannian manifold without boundary, and let \( f \) be a \( {C}^{2} \) function on \( X \) with compact support, such that \( {\Delta f} \geqq 0 \) . Then \( f \) is constant. In particular, every harmonic function with compact support is constant. | Proof. We first give the proof assuming that \( X \) is oriented. By Green’s formula we get\n\n\[ \n{\int }_{X}{\Delta f}{\operatorname{vol}}_{g} = 0 \n\]\n\nSince \( {\Delta f} \geqq 0 \), it follows that in fact \( {\Delta f} = 0 \), so we are reduced to the harmonic case. We now apply Green’s formula to \( {f}^{2} \... | Yes |
Given \( 1 \leqq r \leqq n \), there exists a unique isomorphism\n\n\[ \n* : \mathop{\bigwedge }\limits^{r}V \rightarrow \mathop{\bigwedge }\limits^{{n - r}}V \]\n\n such that for \( \varphi ,\psi \in \mathop{\bigwedge }\limits^{r}V \) we have\n\n\[ \n\langle \varphi ,\psi {\rangle }_{g}{\operatorname{vol}}_{g} = \varp... | The proof will give an explicit determination of the isomorphism on the usual basis for \( \land V \) . Let \( I = \left\lbrack {{i}_{1} < {i}_{2} < \cdots < {i}_{r}}\right\rbrack \) be an ordered set of \( r \) indices. We let\n\n\[ \n{e}_{I} = {e}_{{i}_{1}} \land \cdots \land {e}_{{i}_{r}} \]\n\nIf \( {I}^{\prime } \... | Yes |
Proposition 4.3. The exterior derivative \( d \) has an adjoint \( {d}^{ * } \) with respect to the scalar product \( \langle \rangle ,{\rangle }_{X} \), namely for \( \varphi \in {\mathcal{A}}_{c}^{r - 1}\left( X\right) \) and \( \psi \in {\mathcal{A}}_{c}^{r}\left( X\right) \) we have\n\n\[ \langle {d\varphi },\psi {... | Proof. By Stokes' theorem, we have:\n\n\[ {\int }_{X}{d\varphi } \land * \psi = {\int }_{X}d\left( {\varphi \land * \psi }\right) - {\left( -1\right) }^{r - 1}{\int }_{X} \land d * \psi \]\n\n\[ = {\left( -1\right) }^{r}{\int }_{X}\varphi \land d * \psi \]\n\nNow\n\n\[ {\left( -1\right) }^{r}\varphi \land d * \psi = {\... | Yes |
Lemma 1.1. Let \( \\mathbf{F} \) be a subspace of \( \\mathbf{E} \), let \( x \\in \\mathbf{E} \), and let\n\n\[ a = \\inf \\left| {x - y}\\right| \]\n\nthe inf taken over all \( y \\in \\mathbf{F} \) . Then there exists an element \( {y}_{0} \\in \\mathbf{F} \) such that \( a = \\left| {x - {y}_{0}}\\right| \) . | Proof. Let \( {y}_{n} \) be a sequence in \( \\mathbf{F} \) such that \( \\left| {{y}_{n} - x}\\right| \) tends to \( a \) . We must show that \( {y}_{n} \) is Cauchy. By the parallelogram law,\n\n\[ {\\left| {y}_{n} - {y}_{m}\\right| }^{2} = 2{\\left| {y}_{n} - x\\right| }^{2} + 2{\\left| {y}_{m} - x\\right| }^{2} - 4... | Yes |
Theorem 1.2. If \( \mathbf{F} \) is a subspace properly contained in \( \mathbf{E} \), then there exists a vector \( z \) in \( \mathbf{E} \) which is perpendicular to \( \mathbf{F}\left( {\text{and} \neq 0}\right) \) . | Proof. Let \( x \in \mathbf{E} \) and \( x \notin \mathbf{F} \) . Let \( {y}_{0} \) be an element of \( \mathbf{F} \) which is at minimal distance from \( x \) (use Lemma 1.1). Let \( a \) be this distance and let \( z = {y}_{0} - x \) . After a translation, we may assume that \( z = x \), so that \( \left| x\right| = ... | Yes |
Proposition 2.1. A linear map is bounded if and only if it maps the unit sphere on a bounded subset, if and only if it is continuous. | Proof. Clear. | No |
Proposition 2.2. If \( A \) is an operator and \( \langle {Ax}, x\rangle = 0 \) for all \( x \), then \( A = O \) . | Proof. This follows from the polarization identity,\n\n\[ \langle A\left( {x + y}\right) ,\left( {x + y}\right) \rangle - \langle A\left( {x - y}\right) ,\left( {x - y}\right) \rangle = 2\left\lbrack {\langle {Ax}, y\rangle +\langle {Ay}, x\rangle }\right\rbrack .\n\]\nReplace \( x \) by \( {ix} \) . Then we get\n\n\[ ... | Yes |
Lemma 2.3. Let \( A \) be an operator, and \( {ca} \) number such that\n\n\[ \n\\left| {\\langle {Ax}, x\\rangle }\\right| \\leqq c{\\left| x\\right| }^{2}\n\]\n\nfor all \( x \\in \\mathbf{E} \) . Then for all \( x, y \) we have\n\n\[ \n\\left| {\\langle {Ax}, y\\rangle }\\right| + \\left| {\\langle x,{Ay}\\rangle }\\... | Proof. By the polarization identity,\n\n\[ \n2\\left| {\\langle {Ax}, y\\rangle +\\langle {Ay}, x\\rangle }\\right| \\leqq c{\\left| x + y\\right| }^{2} + c{\\left| x - y\\right| }^{2} = {2c}\\left( {{\\left| x\\right| }^{2} + {\\left| y\\right| }^{2}}\\right) .\n\]\n\nHence\n\n\[ \n\\left| {\\langle {Ax}, y\\rangle +\... | Yes |
Theorem 2.4. We have:\n\n\[ \n{\left( A + B\right) }^{ * } = {A}^{ * } + {B}^{ * },\;{A}^{* * } = A, \]\n\n\[ \n{\left( \alpha A\right) }^{ * } = \bar{\alpha }{A}^{ * },\;\left| {A}^{ * }\right| = \left| A\right| , \]\n\n\[ \n{\left( AB\right) }^{ * } = {B}^{ * }{A}^{ * },\;\left| {A{A}^{ * }}\right| = {\left| A\right|... | Proof. Exercise for the reader. | No |
Proposition 3.1. \( A \) is hermitian if and only if \( \langle {Ax}, x\rangle \) is real for all \( x \) . | Proof. Let \( A \) be hermitian. Then \( \overline{\langle {Ax}, x\rangle } = \overline{\langle x,{Ax}\rangle } = \langle {Ax}, x\rangle \) . Conversely, \( \langle {Ax}, x\rangle = \overline{\langle {Ax}, x\rangle } = \langle x,{Ax}\rangle = \left\langle {{A}^{ * }x, x}\right\rangle \) implies that\n\n\[\n\left\langle... | Yes |
Proposition 3.2. Let \( A \) be a hermitian operator. Then \( \left| A\right| \) is the greatest lower bound of all values \( c \) such that\n\n\[ \left| {\langle {Ax}, x\rangle }\right| \leqq c{\left| x\right| }^{2} \]\n\nfor all \( x \), or equivalently, the sup of all values \( \left| {\langle {Ax}, x\rangle }\right... | Proof. When \( A \) is hermitian we obtain\n\n\[ \left| {\langle {Ax}, y\rangle }\right| \leqq c\left| x\right| \left| y\right| \]\n\nfor all \( x, y \in E \), so that we get \( \left| A\right| \leqq c \) in Lemma 2.3. On the other hand, \( c = \left| A\right| \) is certainly a possible value for \( c \) by the Schwart... | No |
Theorem 3.3. Let \( \alpha ,\beta \) be real and \( {\alpha I} \leqq A \leqq {\beta I} \) . Let \( p \) be a real polynomial, semipositive in the interval \( \alpha \leqq t \leqq \beta \) . Then \( p\left( A\right) \) is a semipositive operator. | Proof. We shall need the following obvious facts.\n\nIf \( A, B \) are hermitian, \( A \) commutes with \( B \), and \( A \geqq O \), then \( A{B}^{2} \) is semipositive.\n\nIf \( p\left( t\right) \) is quadratic, of type \( p\left( t\right) = {t}^{2} + {at} + b \) and has imaginary roots, then\n\n\[ p\left( t\right) =... | Yes |
Corollary 3.5. Let \( {\alpha I} \leqq A \leqq {\beta I} \) . Let \( p\left( t\right) \) be a real polynomial. Then \( \left| {p\left( A\right) }\right| \leqq \parallel p\parallel \) | Proof. Let \( q\left( t\right) = \parallel p\parallel \pm p\left( t\right) \) . Then \( q\left( t\right) \) is \( \geqq 0 \) on the interval. Hence \( q\left( A\right) \geqq O \) and our assertion follows at once. | Yes |
Corollary 3.5. Let \( {\alpha I} \leqq A \leqq {\beta I} \) . Let \( p\left( t\right) \) be a real polynomial. Then \( \left| {p\left( A\right) }\right| \leqq \parallel p\parallel \) | Proof. Let \( q\left( t\right) = \parallel p\parallel \pm p\left( t\right) \) . Then \( q\left( t\right) \) is \( \geqq 0 \) on the interval. Hence \( q\left( A\right) \geqq O \) and our assertion follows at once. | No |
Proposition 3.6. Let \( A \) be a semipositive operator. Then there exists an operator \( B \) in the closure of the algebra generated by \( A \) such that \( {B}^{2} = A \) . | Proof. The continuous function \( {t}^{1/2} \) maps on \( {A}^{1/2} \) . | No |
Corollary 3.7. The product of two semipositive, commuting hermitian operators is again semipositive. | Proof. Let \( A, C \) be hermitian and \( {AC} = {CA} \) . If \( B \) is as in Proposition 3.6 , then\n\n\[ \langle {ACx}, x\rangle = \left\langle {{B}^{2}{Cx}, x}\right\rangle = \langle {BCx},{Bx}\rangle = \langle {CBx},{Bx}\rangle \geqq 0. \] | Yes |
Lemma 3.8. Let \( X \) be a compact set, \( R \) the ring of continuous functions on \( X \), and \( \mathfrak{a} \) a closed ideal of \( R,\mathfrak{a} \neq R \) . Let \( C \) be the closed set of zeros of \( \mathfrak{a} \) . Then \( C \) is not empty and if a function \( f \in R \) vanishes on \( C \), then \( f \in... | Proof. Given \( \epsilon \), let \( U \) be the open set where \( \left| f\right| < \epsilon \) . Then \( X - U \) is closed. For each point \( t \in X - U \) there exists a function \( g \in \mathfrak{a} \) such that \( g\left( t\right) \neq 0 \) in a neighborhood of \( t \) . These neighborhoods cover \( X - U \), an... | No |
Theorem 3.9. The map\n\n\\[ \nf\\left( t\\right) \\mapsto f\\left( A\\right) \n\\]\n\ninduces a Banach-isomorphism (i.e. norm-preserving) of the Banach algebra of continuous functions on \\( \\sigma \\left( A\\right) \\) onto the closure of the algebra generated by \\( A \\) . | Proof. We have already proved that our map is an agebraic isomorphism and that \\( \\left| {f\\left( A\\right) }\\right| \\leqq \\parallel f{\\parallel }_{A} \\) . In order to get the reverse inequality, we shall prove:\n\nIf \\( f\\left( A\\right) \\geqq O \\), then \\( f\\left( t\\right) \\geqq 0 \\) on the spectrum ... | Yes |
Theorem 3.10. The general spectrum is compact, and in fact, if \( \xi \) is in it, then \( \left| \xi \right| \leqq \left| A\right| \) . If \( A \) is hermitian, then the general spectrum is equal to \( \sigma \left( A\right) \) . | Proof. The complement of the general spectrum is open, because if \( A - {\xi }_{0} \) is invertible, and \( \xi \) is close to \( {\xi }_{0} \), then \( {\left( A - {\xi }_{0}\right) }^{-1}\left( {A - \xi }\right) \) is close to \( I \), hence invertible, and hence \( A - \xi \) is also invertible. Furthermore, if \( ... | Yes |
Theorem 3.11. Let \( S \) be a set of operators of the Hilbert space \( E \), leaving no closed subspace invariant except 0 and \( \mathbf{E} \) itself. Let \( A \) be a Hermitian operator such that \( {AB} = {BA} \) for all \( B \in S \) . Then \( A = {\lambda I} \) for some real number \( \lambda \) . | Proof. It will suffice to prove that there is only one element in the spectrum of \( A \) . Suppose there are two, \( {\lambda }_{1} \neq {\lambda }_{2} \) . There exist continuous functions \( f, g \) on the spectrum such that neither is 0 on the spectrum, but \( {fg} \) is 0 on the spectrum. For instance, one may tak... | Yes |
Corollary 3.12. Let \( S \) be a set of operators of the Hilbert space \( \mathbf{E} \) , leaving no closed subspace invariant except 0 and \( \mathbf{E} \) itself. Let \( A \) be an operator such that \( A{A}^{ * } = {A}^{ * }A,{AB} = {BA} \), and \( {A}^{ * }B = B{A}^{ * } \) for all \( B \in S \) . Then \( A = {\lam... | Proof. Write \( A = {A}_{1} + i{A}_{2} \) where \( {A}_{1},{A}_{2} \) are hermitian and commute (e.g. \( {A}_{1} = \left( {A + {A}^{ * }}\right) /2 \) ). Apply the theorem to each one of \( {A}_{1} \) and \( {A}_{2} \) to get the result. | No |
Proposition 1.1 A nonempty set \( X \) is countable if and only if there is a surjection from \( \mathbb{N} \) onto \( X \) . | Proof. If \( X \) is countably infinite there is a bijection, and thus a surjection, from \( \mathbb{N} \) to \( X \) . If \( X \) is finite with \( n \geq 1 \) elements, there is a bijection \( \varphi : \{ 1,\ldots, n\} \rightarrow X \) . This can be arbitrarily extended to a bijection from \( \mathbb{N} \) to \( X \... | Yes |
Corollary 1.2 If \( X \) is countable and there exists a surjection from \( X \) to \( Y \), then \( Y \) is countable. | Indeed, the composition of two surjections is surjective. | No |
Corollary 1.4 If \( Y \) is countable and there exists an injection from \( X \) to \( Y \), then \( X \) is countable. | Proof. An injection \( f : X \rightarrow Y \) defines a bijection from \( X \) to \( f\left( X\right) \). If \( Y \) is countable, so is \( f\left( X\right) \), by the preceding corollary. Therefore \( X \) is countable. | Yes |
Corollary 1.4 If \( Y \) is countable and there exists an injection from \( X \) to \( Y \), then \( X \) is countable. | Proof. An injection \( f : X \rightarrow Y \) defines a bijection from \( X \) to \( f\left( X\right) \) . If \( Y \) is countable, so is \( f\left( X\right) \), by the preceding corollary. Therefore \( X \) is countable. | Yes |
Proposition 1.6 If the sets \( {X}_{1},{X}_{2},\ldots ,{X}_{n} \) are countable, the Cartesian product \( X = {X}_{1} \times {X}_{2} \times \cdots \times {X}_{n} \) is countable. | Proof. It is enough to prove the result for \( n = 2 \) and use induction. Suppose that \( {X}_{1} \) and \( {X}_{2} \) are countable, and let \( {f}_{1},{f}_{2} \) be surjections from \( \mathbb{N} \) to \( {X}_{1},{X}_{2} \) (whose existence is given by Proposition 1.1). The map \( \left( {{n}_{1},{n}_{2}}\right) \ma... | Yes |
Proposition 1.7 Let \( {\left( {X}_{i}\right) }_{i \in I} \) be a family of countable sets, indexed by a countable set \( I \) . The set \( X = \mathop{\bigcup }\limits_{{i \in I}}{X}_{i} \) is countable. | Proof. If, for each \( i \in I \), we take a surjection \( {f}_{i} : \mathbb{N} \rightarrow {X}_{i} \), the map \( f : I \times \mathbb{N} \rightarrow X \) defined by \( f\left( {i, n}\right) = {f}_{i}\left( n\right) \) is a surjection. But \( I \times \mathbb{N} \) is countable. | Yes |
Proposition 2.1 Every compact metric space is separable. | Proof. If \( n \) is a strictly positive integer, the union of the balls \( B\left( {x,\frac{1}{n}}\right) \) , over \( x \in X \), covers \( X \) . By the Borel-Lebesgue property, \( X \) can be covered by a finite number of such balls: \( X = \mathop{\bigcup }\limits_{{j = 1}}^{{J}_{n}}B\left( {{x}_{j}^{n},\frac{1}{n... | Yes |
Proposition 2.2 Every \( \sigma \) -compact metric space is separable. | This is an immediate consequence of Propositions 2.1 and 1.7. | No |
Proposition 2.3 If \( X \) is a separable metric space and \( Y \) is a subset of \( X \), then \( Y \) is separable (in the induced metric). | Proof. Let \( \left( {x}_{n}\right) \) be a dense sequence in \( X \) . Set\n\n\[ \mathcal{U} = \left\{ {\left( {n, p}\right) \in \mathbb{N} \times {\mathbb{N}}^{ * } : B\left( {{x}_{n},1/p}\right) \cap Y \neq \varnothing }\right\} .\n\]\n\nFor each \( \left( {n, p}\right) \in \mathcal{U} \), choose a point \( {x}_{n, ... | Yes |
Proposition 2.5 A normed space is separable if and only if it contains a countable fundamental family of vectors. | Proof. The condition is certainly necessary, since a dense family of vectors is fundamental. Conversely, let \( D \) be a countable fundamental family of vectors in a normed space \( E \) . Let \( \mathcal{D} \) be the set of linear combinations of elements of \( D \) with coefficients in the field \( Q = \mathbb{Q} \)... | Yes |
Proposition 2.6 A normed space is separable if and only if it has a countable topological basis. | Proof. The \ | No |
Theorem 3.1 Let \( {\left( {X}_{p},{d}_{p}\right) }_{p \in \mathbb{N}} \) be a sequence of metric spaces, and, for every \( p \in \mathbb{N} \), let \( {\left( {x}_{n, p}\right) }_{n \in \mathbb{N}} \) be a sequence in \( {X}_{p} \) . If, for every \( p \in \mathbb{N} \), the set \( \left\{ {{x}_{n, p} : n \in \mathbb{... | Proof. Thanks to the assumption of relative compactness, one can inductively construct a decreasing subsequence \( \left( {A}_{n}\right) \) of infinite subsets of \( \mathbb{N} \) such that, for every \( p \in \mathbb{N} \), the sequence \( {\left( {x}_{n, p}\right) }_{n \in {A}_{p}} \) converges in \( {X}_{p} \) . The... | Yes |
Corollary 3.2 (Tychonoff’s Theorem) If \( {\left( {X}_{p}\right) }_{p \in \mathbb{N}} \) is a sequence of compact metric spaces and \( X = \mathop{\prod }\limits_{{p \in \mathbb{N}}}{X}_{p} \) is the product space (with the product distance), \( X \) is compact. | This follows immediately from the definition of the product metric, from Theorem 3.1, and from the characterization of compact sets by the Bolzano-Weierstrass property. | No |
Theorem 3.3 Let \( X \) be a metric space. Every relatively compact subset of \( X \) is precompact. The converse is true if \( X \) is complete. | Proof. The first statement follows directly from the definitions, from the Borel-Lebesgue property of compact sets, and from the fact that \( A \subset X \) implies \( \bar{A} \subset \mathop{\bigcup }\limits_{{x \in X}}B\left( {x,\varepsilon }\right) \) for every \( \varepsilon > 0 \) .\n\nNow suppose that \( X \) is ... | Yes |
Proposition 4.1 Consider a normed space \( E \), a fundamental family \( D \) in \( E \), and a Banach space \( F \) . Consider also a bounded sequence \( {\left( {T}_{n}\right) }_{n \in \mathbb{N}} \) of elements of \( L\left( {E, F}\right) \) . If, for every \( x \in D \), the sequence \( {\left( {T}_{n}x\right) }_{n... | Proof. Let \( M > 0 \) be such that \( \begin{Vmatrix}{T}_{n}\end{Vmatrix} \leq M \) for all \( n \in \mathbb{N} \) . It is clear that the sequence \( \left( {{T}_{n}x}\right) \) converges for any element \( x \) of the vector space \( \left\lbrack D\right\rbrack \) generated by \( D \) . Now take \( x \in E \) and \( ... | Yes |
Proposition 4.3 Consider normed spaces \( E \) and \( F \), a fundamental set \( D \) in \( E \), a bounded sequence \( \left( {T}_{n}\right) \) in \( L\left( {E, F}\right) \) and a map \( T \in L\left( {E, F}\right) \) . If the sequence \( \left( {{T}_{n}x}\right) \) converges toward \( {Tx} \) for every point \( x \i... | Proof. By taking differences we can suppose that \( T = 0 \) . Set\n\n\[ M = \mathop{\sup }\limits_{{n \in \mathbb{N}}}\begin{Vmatrix}{T}_{n}\end{Vmatrix} \]\n\nand take \( x \in E \) . For every \( y \in \left\lbrack D\right\rbrack \), we have\n\n\[ \begin{Vmatrix}{{T}_{n}x}\end{Vmatrix} \leq M\parallel x - y\parallel... | Yes |
Proposition 1.1 \( C\left( X\right) \) is a separable Banach space. | Proof. The reader can check that \( C\left( X\right) \) is a Banach space. We show separability. Since \( X \) is precompact, for every \( n \in {\mathbb{N}}^{ * } \) there exist finitely many points \( {x}_{1}^{n},\ldots ,{x}_{{N}_{n}}^{n} \) of \( X \) such that \( X = \mathop{\bigcup }\limits_{{j = 1}}^{{N}_{n}}B\le... | Yes |
Proposition 1.2 (Dini’s Lemma) \( \operatorname{Let}{\left( {f}_{n}\right) }_{n \in \mathbb{N}} \) be an increasing sequence in \( {C}^{\mathbb{R}}\left( X\right) \) (this means that \( {f}_{n} \leq {f}_{n + 1} \) for all \( n \) ). If the sequence \( \left( {f}_{n}\right) \) converges pointwise to a function \( f \in ... | Proof. Take \( \varepsilon > 0 \) . For every \( n \in \mathbb{N} \) we set \( {\Omega }_{n} = \left\{ {x \in X : {f}_{n}\left( x\right) > }\right. \) \( f\left( x\right) - \varepsilon \} \) . Clearly, \( \left( {\Omega }_{n}\right) \) is an increasing sequence of open subsets in \( X \) whose union is \( X \) . By the... | Yes |
Lemma 2.1 Suppose \( X \) has at least two elements. Let \( H \) be a subset of \( {C}^{\mathbb{R}}\left( X\right) \) satisfying these two conditions:\n\na. For all \( u, v \in H \), the functions \( \sup \left( {u, v}\right) \) and \( \inf \left( {u, v}\right) \) also lie in \( H \) .\n\nb. If \( {x}_{1},{x}_{2} \) ar... | Proof. Take \( f \in {C}^{\mathbb{R}}\left( X\right) \) and \( \varepsilon > 0 \) . We want to find an element of \( H \) that is \( \varepsilon \) -close to \( f \) . First fix \( x \in X \) . By assumption \( \mathrm{b} \), for every \( y \neq x \) there exists \( {u}_{y} \in H \) such that \( {u}_{y}\left( x\right) ... | Yes |
Theorem 2.2 If \( H \) is a separating vector subspace of \( {C}^{\mathbb{R}}\left( X\right) \) that is a lattice and contains the constants, then \( H \) is dense in \( {C}^{\mathbb{R}}\left( X\right) \) . | Proof. If \( X \) has a single element, the result is clear. Suppose \( X \) has at least two elements; we just need to check assumption b of the lemma. Let \( {x}_{1} \) and \( {x}_{2} \) be distinct elements of \( X \) . Since \( H \) is separating, there exists \( h \in H \) such that \( h\left( {x}_{1}\right) \neq ... | Yes |
Theorem 2.3 (Stone-Weierstrass Theorem, real case) Every separating subalgebra of \( {C}^{\mathbb{R}}\left( X\right) \) containing the constant functions is dense in \( {C}^{\mathbb{R}}\left( X\right) \) . | Proof. If \( H \) is a separating subalgebra of \( {C}^{\mathbb{R}}\left( X\right) \) containing the constants, so is its closure \( \bar{H} \) . Therefore it suffices to show that \( \bar{H} \) is a lattice and to apply Theorem 2.2. Thus, let \( f \) be a nonzero element of \( \bar{H} \) . We saw in the example on pag... | Yes |
Theorem 2.4 (Stone–Weierstrass Theorem, complex case) Every separating subalgebra \( H \) of \( {C}^{\mathbb{C}}\left( X\right) \) that is self-conjugate and contains the constant functions is dense in \( {C}^{\mathbb{C}}\left( X\right) \) . | Proof. Set \( {H}_{\mathbb{R}} = \{ h \in H : h\left( x\right) \in \mathbb{R} \) for all \( x \in X\} \) . Clearly, \( {H}_{\mathbb{R}} \) is a subalgebra of \( {C}^{\mathbb{R}}\left( X\right) \) containing the constants. Now, if \( f \in H \), the real and imaginary parts of \( f \) lie in \( {H}_{\mathbb{R}} \), sinc... | Yes |
Lemma 2.5 The map from \( {C}^{\mathbb{C}}\left( \mathbb{U}\right) \) to \( {C}_{2\pi }^{\mathbb{C}} \) that associates to \( \varphi \in \) \( {C}^{\mathbb{C}}\left( \mathbb{U}\right) \) the function \( f \) given by \( f\left( \theta \right) = \varphi \left( {e}^{i\theta }\right) \) for every real \( \theta \) is a s... | Proof. Only the surjectivity requires proof. For \( z \in \mathbb{U} \), denote by \( \arg z \) some real number such that \( {e}^{i\arg z} = z \) . We know that \( \arg z \) is defined modulo \( {2\pi } \) and that there exist choices of \( \arg z \) that vary continuously in the neighborhood of a given point (for exa... | Yes |
Proposition 3.1 A subset of \( C\left( X\right) \) is equicontinuous if and only if it is uniformly equicontinuous. | Proof. It is enough to show necessity. Let \( H \) be an equicontinuous subset of \( C\left( X\right) \), and let \( \varepsilon > 0 \) be a real number. By assumption, for every \( x \in X \) there exists \( {\eta }_{x} > 0 \) such that \( \left| {h\left( y\right) - h\left( x\right) }\right| < \varepsilon /2 \) whenev... | Yes |
Proposition 3.2 Let \( \left( {f}_{n}\right) \) be an equicontinuous sequence in \( C\left( X\right) \) and let \( D \) be a dense subset of \( X \) . If, for all \( x \in D \), the sequence of numbers \( \left( {{f}_{n}\left( x\right) }\right) \) converges, the sequence of functions \( \left( {f}_{n}\right) \) converg... | Proof. It suffices to show that \( \left( {f}_{n}\right) \) is a Cauchy sequence in \( C\left( X\right) \) . To do this, take \( \varepsilon > 0 \) . By assumption, there exists \( \eta > 0 \) such that, whenever \( d\left( {x, y}\right) < \eta , \)\n\n\[ \left| {{f}_{n}\left( x\right) - {f}_{n}\left( y\right) }\right|... | Yes |
Theorem 3.3 (Ascoli) A subset of \( C\left( X\right) \) is relatively compact in \( C\left( X\right) \) if and only if it is bounded and equicontinuous. | Proof. For the \ | No |
Proposition 3.4 The image under \( T \) of the closed unit ball of \( C\left( Y\right) \) is a relatively compact subset of \( C\left( X\right) \) . | Proof. It is clear that \( T\left( {\bar{B}\left( {C\left( Y\right) }\right) }\right) \) is bounded by\n\n\[ M = \mu \left( Y\right) \mathop{\max }\limits_{{\left( {x, y}\right) \in X \times Y}}\left| {K\left( {x, y}\right) }\right| .\n\]\n\nOn the other hand, \( K \) is uniformly continuous on \( X \times Y \) ; in pa... | Yes |
Theorem 1.1 (F. Riesz) Let \( X \) be a normed space, with open unit ball \( B \) and closed unit ball \( \bar{B} \). The following properties are equivalent:\n\ni. \( X \) is finite-dimensional.\n\nii. \( X \) is locally compact.\n\niii. \( \bar{B} \) is compact.\n\niv. \( B \) is precompact. | Proof. Property i implies ii because closed balls in a finite-dimensional normed space are compact. If ii is true, there exists \( r > 0 \) such that \( \bar{B}\left( {0, r}\right) = r\bar{B} \) is compact; this implies iii. That iii implies iv is obvious. Thus the only nontrivial part of the theorem is iv \( \Rightarr... | Yes |
Proposition 1.2 If \( X \) is a locally compact space, there exists for every \( x \in X \) and for every neighborhood \( V \) of \( x \) a real number \( r > 0 \) such that \( \bar{B}\left( {x, r}\right) \) is compact and \( \bar{B}\left( {x, r}\right) \subset V \) . | Proof. Just choose \( r = \min \left( {{r}^{\prime },{r}^{\prime \prime }}\right) \), where \( {r}^{\prime } \) and \( {r}^{\prime \prime } \) are such that \( \bar{B}\left( {x,{r}^{\prime }}\right) \) is compact and \( \bar{B}\left( {x,{r}^{\prime \prime }}\right) \subset V \) . | Yes |
Corollary 1.3 Let \( X \) be locally compact. If \( O \) is open in \( X \) and \( F \) is closed in \( X \), the intersection \( Y = O \cap F \) (with the induced metric) is locally compact. | Proof. Take \( x \in Y \) . By the preceding proposition, there exists \( r > 0 \) such that \( \bar{B}\left( {x, r}\right) \) is compact and contained in \( O \) . Then \( \bar{B}\left( {x, r}\right) \cap Y = \bar{B}\left( {x, r}\right) \cap F \) is compact. | Yes |
Corollary 1.4 Consider a locally compact space \( X \), a compact subset \( K \) of \( X \), and open subsets \( {O}_{1},\ldots ,{O}_{n} \) of \( X \) covering \( K \) . There exist compact sets \( {K}_{1},\ldots ,{K}_{n} \) with \( {K}_{j} \subset {O}_{j} \) for each \( j \) and such that\n\n\[ K \subset \mathop{\bigc... | Proof. By Proposition 1.2, for all points \( x \) of \( K \) there exists \( j \in \{ 1,\ldots, n\} \) and a compact set \( {K}_{x} \) such that \( x \in {\mathring{K}}_{x} \subset {K}_{x} \subset {O}_{j} \) . By the Borel-Lebesgue property, \( K \) can be covered by finitely many of these interiors:\n\n\[ K \subset \m... | Yes |
Proposition 1.5 Let \( X \) be a locally compact space. The following properties are equivalent:\n\ni. \( X \) is separable.\n\nii. \( X \) is \( \sigma \) -compact.\n\niii. There exists a sequence \( \left( {K}_{n}\right) \) of compact sets covering \( X \) and such that \( {K}_{n} \subset {\overset{ \circ }{K}}_{n + ... | Proof. It is clear that iii implies ii. The implication ii \( \Rightarrow \) i is a particular case of Proposition 2.2 on page 8.\n\nNow suppose that \( X \) is separable and let \( \left( {x}_{n}\right) \) be a sequence dense in \( X \) . Set \( A = \left\{ {\left( {n, p}\right) \in \mathbb{N} \times {\mathbb{N}}^{ * ... | Yes |
Proposition 1.6 Let \( \left( {K}_{n}\right) \) be a sequence of compact sets that exhausts a metric space \( X \) . For every compact \( K \) of \( X \) there exists an integer \( n \) such that \( K \subset {K}_{n} \) . | Proof. The open sets \( {\mathring{K}}_{n} \) cover \( K \) . By the Borel-Lebesgue property, \( K \) is in fact contained in a finite union of sets \( {\mathring{K}}_{n} \) : but \( \mathop{\bigcup }\limits_{{j \leq n}}{\mathring{K}}_{j} = {\mathring{K}}_{n} \) . | Yes |
Proposition 1.7 Let \( {\left( {f}_{n}\right) }_{n \in \mathbb{N}} \) be an increasing sequence in \( {C}_{0}^{\mathbb{R}}\left( X\right) \), converging pointwise to a function \( f \in {C}_{0}^{\mathbb{R}}\left( X\right) \) . Then \( \left( {f}_{n}\right) \) converges uniformly to \( f \) . | Proof. We show that the sequence \( \left( {g}_{n}\right) \) defined by \( {g}_{n} = f - {f}_{n} \) converges uniformly to 0 . Given \( \varepsilon > 0 \), there exists a compact \( K \) such that \( {g}_{0}\left( x\right) \leq \varepsilon \) for all \( x \notin K \) . By Dini’s Lemma, there exists an integer \( n \) s... | Yes |
Proposition 1.8 (Partitions of unity) Let \( X \) be locally compact. If \( K \) is a compact subset of \( X \) and \( {O}_{1},\ldots ,{O}_{n} \) are open subsets of \( X \) that cover \( K \), there exist functions \( {\varphi }_{1},\ldots ,{\varphi }_{n} \) in \( {C}_{c}^{\mathbb{R}}\left( X\right) \) such that \( 0 ... | Proof. Let \( {K}_{1},\ldots ,{K}_{n} \) be the compact sets whose existence is granted by Corollary 1.4. We just have to set, for \( x \in X \) ,\n\n\[ {\varphi }_{j}\left( x\right) = \frac{d\left( {x, X \smallsetminus {\mathring{K}}_{j}}\right) }{d\left( {x, K}\right) + \mathop{\sum }\limits_{{k = 1}}^{n}d\left( {x, ... | Yes |
Corollary 1.9 If \( X \) is locally compact, \( {C}_{c}\left( X\right) \) is dense in \( {C}_{0}\left( X\right) \) . | Proof. Take \( f \in {C}_{0}\left( X\right) \) and \( \varepsilon > 0 \) . Let \( K \) be a compact such that \( \left| {f\left( x\right) }\right| < \varepsilon \) for all \( x \notin K \) . Applying Proposition 1.8 with \( n = 1 \) and \( {O}_{1} = X \), we find a \( \varphi \in {C}_{c}^{\mathbb{R}}\left( X\right) \) ... | Yes |
Corollary 1.10 Let \( X \) be locally compact and separable and let \( O \) be open in \( X \) . There exists an increasing sequence \( \left( {\varphi }_{n}\right) \) of functions in \( {C}_{c}^{ + }\left( X\right) \), each with support contained in \( O \), and such that \( \mathop{\lim }\limits_{{n \rightarrow + \in... | Proof. \( O \) is a locally compact separable space, by Corollary 1.3 above and Proposition 2.3 on page 8. By Proposition 1.5 there exists a sequence of compact sets \( \left( {K}_{n}\right) \) such that \( {K}_{n} \subset {\mathring{K}}_{n + 1} \) for all \( n \) and \( \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{K}... | Yes |
Lemma 2.1 \( \mathcal{L} \) is the smallest subset of \( \mathcal{F} \) that contains \( L \) and is closed under pointwise convergence (the latter condition means that the pointwise limit of any sequence in \( \mathcal{L} \) is also in \( \mathcal{L} \) ). | Proof. It is clear that a minimal set satisfying these conditions exists. Call it \( \mathcal{B} \). - \( \mathcal{B} \) is a vector subspace of \( \mathcal{F} \) and a lattice, and it contains the constants. Proof. If \( \lambda \in \mathbb{R} \), the set \( \{ f \in \mathcal{F} : {\lambda f} \in \mathcal{B}\} \) cont... | Yes |
Proposition 2.2 If \( X \) is a metric space, the set of Borel functions from \( X \) to \( \mathbb{R} \) is the smallest subset of \( \mathcal{F} \) that contains all continuous functions from \( X \) to \( \mathbb{R} \) and is closed under pointwise convergence. | Proof. Let \( L \) be the set of continuous functions from \( X \) to \( \mathbb{R} \) . Then \( L \) is a lattice and satisfies \( \left( *\right) \), since \( 1 \in L \) . On the other hand, let \( \mathcal{B} \) be the Borel \( \sigma \) -algebra of \( X \) . Certainly every continuous function on \( X \) is \( \mat... | Yes |
Theorem 2.3 (Daniell) Let \( \mu \) be a linear form on \( L \) satisfying these conditions:\n\n1. \( \mu \) is positive, that is, if \( f \in L \) satisfies \( f \geq 0 \) then \( \mu \left( f\right) \geq 0 \) .\n\n2. If a sequence \( \left( {f}_{n}\right) \) in \( L \) satisfies \( {f}_{n} \searrow 0 \), then \( \mat... | Uniqueness of \( m \) . Suppose that two measures \( {m}_{1} \) and \( {m}_{2} \) satisfy the stated properties. Let \( \left( {\varphi }_{n}\right) \) be a sequence satisfying condition \( \left( *\right) \) on page 58 . For every \( n \in \mathbb{N} \) and every real \( \lambda \geq 0 \), the set\n\n\[ \left\{ {f \in... | Yes |
Proposition 2.4 Under the same assumptions and with the same notation as in Theorem 2.3, the space \( L \) is dense in the Banach space \( {L}^{1}\left( m\right) \) . | Proof. We maintain the same notation. It suffices to show that if \( A \) is in \( \sigma \left( L\right) \) and \( m\left( A\right) \) is finite then for every \( \varepsilon > 0 \) there exists an element \( \varphi \) of \( L \) such that \( \mu \left( \left| {{1}_{A} - \varphi }\right| \right) < \varepsilon \) . If... | No |
Proposition 3.1 Let \( m \) be a Borel measure on \( X \) . There exists a largest open set \( O \) such that \( m\left( O\right) = 0 \) . | Proof. Let \( \mathcal{U} \) be the set of all open sets \( \Omega \) of \( X \) such that \( m\left( \Omega \right) = 0 \) . This set is nonempty since it contains \( \varnothing \) . Set \( O = \mathop{\bigcup }\limits_{{\Omega \in \mathcal{U}}}\Omega \) ; this is an open set, which we must prove has \( m \) -measure... | Yes |
Proposition 3.3 Let \( \mu \) be a positive Radon measure on \( X \) . For every compact set \( K \) in \( X \), the restriction of \( \mu \) to \( {C}_{K}^{\mathbb{R}}\left( X\right) \) is continuous; that is, there exists a constant \( {C}_{K} \geq 0 \) such that\n\n\[ \left| {\mu \left( f\right) }\right| \leq {C}_{K... | Proof. Let \( K \) be compact in \( X \) . By Proposition 1.8 on page 53, there exists \( {\varphi }_{K} \in {C}_{c}^{ + }\left( X\right) \) such that \( 0 \leq {\varphi }_{K} \leq 1 \) and \( {\varphi }_{K} = 1 \) on \( K \) . Then, for all \( f \in {C}_{K}^{\mathbb{R}}\left( X\right) \), we have \( \left| f\right| \l... | Yes |
Theorem 3.4 (Radon-Riesz) For every positive Radon measure \( \mu \) on \( X \) there exists a unique Borel measure \( m \) finite on compact sets and such that \[ \mu \left( f\right) = \int {fdm}\;\text{ for all }f \in {C}_{c}^{\mathbb{R}}\left( X\right) . \] The map \( \mu \mapsto m \) thus defined is a bijection bet... | Proof. This will follow as a particular case of Daniell's Theorem. Set \( L = {C}_{c}^{\mathbb{R}}\left( X\right) \) . This space satisfies the assumptions stated on page 58 : in particular, property (*) follows from Corollary 1.10 on page 53. Now take \( \mu \in {\mathfrak{M}}^{ + }\left( X\right) \) ; we will show th... | Yes |
Proposition 3.6 For every positive linear form \( \mathfrak{m} \) on \( {C}_{0}^{\mathbb{R}}\left( X\right) \) there exists a unique positive Radon measure \( \mu \) of finite mass and such that \( \mathfrak{m} = {\mathfrak{m}}_{\mu } \) , or equivalently such that\n\n\[ \mathfrak{m}\left( f\right) = \int {fd\mu }\;\te... | Proof. The uniqueness of \( \mu \) clearly follows from the inclusion of \( {C}_{c}^{\mathbb{R}}\left( X\right) \) in \( {C}_{0}^{\mathbb{R}}\left( X\right) \) . The important point is existence.\n\nWe first show that \( \mathfrak{m} \) is continuous. If not, there exists a sequence \( \left( {f}_{n}\right) \) in \( {C... | Yes |
Lemma 3.7 Let \( \alpha \) be an increasing function from \( \mathbb{R} \) to \( \mathbb{R} \) . If \( a \) and \( b \) are real numbers with \( a < b \), then\n\n\[ \n{d\alpha }(\left( {a, b\rbrack }\right) = \alpha \left( {b}_{ + }\right) - \alpha \left( {a}_{ + }\right)\n\]\n\nwhere \( \alpha \left( {a}_{ + }\right)... | Proof. Let \( {\left( {\varphi }_{n}\right) }_{n \geq 1} \) be a sequence in \( {C}_{c}^{\mathbb{R}}\left( X\right) \) such that \( 0 \leq {\varphi }_{n} \leq 1,{\varphi }_{n} = 1 \) on \( \left\lbrack {a + 1/n, b - 1/n}\right\rbrack \), and \( {\varphi }_{n} = 0 \) on \( \mathbb{R} \smallsetminus \left\lbrack {a + 1/\... | Yes |
Theorem 3.8 Let \( \mu \) be a positive Radon measure on \( \mathbb{R} \) . There exists a unique increasing right-continuous function \( \alpha \) with \( \alpha \left( 0\right) = 0 \) and \( \mu = {d\alpha } . | Proof. Uniqueness is clear since, by the preceding discussion, if \( \alpha \) is right-continuous and vanishes at 0 , it is determined everywhere:\n\n\[ \alpha \left( x\right) = \left\{ \begin{array}{ll} - \mu \left( {(x,0\rbrack }\right) & \text{ if }x < 0 \\ 0 & \text{ if }x = 0 \\ \mu \left( {(0, x\rbrack }\right) ... | Yes |
Every bounded real Radon measure is the difference of two positive Radon measures of finite mass. More precisely, if \( \mu \in {\mathfrak{M}}_{f}^{\mathbb{R}}\left( X\right) \) , the Radon measures \( {\mu }^{ + } \) and \( {\mu }^{ - } \) defined in Theorem 4.1 have finite mass and\n\n\[ \parallel \mu \parallel = \in... | Proof. We first see that, for any \( f \in {C}_{c}^{ + }\left( X\right) \), \n\n\[ {\mu }^{ + }\left( f\right) + {\mu }^{ - }\left( f\right) = \sup \left\{ {\mu \left( {g - h}\right) : g, h \in {C}_{c}^{ + }\left( X\right) \text{ and }g, h \leq f}\right\} \]\n\n\[ = \sup \left\{ {\mu \left( \varphi \right) : \varphi \i... | Yes |
Proposition 1.1 (Schwarz inequality) Let \( E \) be a vector space with a scalar semiproduct \( \left( {\cdot \mid \cdot }\right) \) . For every \( x, y \in E \) , \[ {\left| \left( x \mid y\right) \right| }^{2} \leq \left( {x \mid x}\right) \left( {y \mid y}\right) \] | Proof. One can assume \( \mathbb{K} = \mathbb{C} \) . If \( x, y \in E \) , \( \left( {x + {ty} \mid x + {ty}}\right) = \left( {x \mid x}\right) + {2t}\operatorname{Re}\left( {x \mid y}\right) + {t}^{2}\left( {y \mid y}\right) \geq 0\; \) for all \( t \in \mathbb{R}. \) Consider the expression on the left-hand side of ... | Yes |
Corollary 1.2 Let \( E \) be a vector space with a scalar product \( \left( {\cdot \mid \cdot }\right) \) . The expression \( \parallel x\parallel = {\left( x \mid x\right) }^{1/2} \) defines a norm on \( E \) . | Proof. It is enough to check the triangle inequality. We have\n\n\[ \parallel x + y{\parallel }^{2} = \parallel x{\parallel }^{2} + \parallel y{\parallel }^{2} + 2\operatorname{Re}\left( {x \mid y}\right) \]\n\n\[ \leq \parallel x{\parallel }^{2} + \parallel y{\parallel }^{2} + 2\parallel x\parallel \parallel y\paralle... | Yes |
Corollary 1.3 Let \( E \) be a scalar product space. For every \( y \in E \), the linear form \( {\varphi }_{y} = \left( {\cdot \mid y}\right) \) is continuous and its norm in the topological dual \( {E}^{\prime } \) of \( E \) equals \( \parallel y\parallel \) . | Proof. By the Schwarz inequality, \( \left| {{\varphi }_{y}\left( x\right) }\right| \leq \parallel x\parallel \parallel y\parallel \) for all \( x \in E \), so \( {\varphi }_{y} \in \) \( {E}^{\prime } \) and \( \begin{Vmatrix}{\varphi }_{y}\end{Vmatrix} \leq \parallel y\parallel \) . At the same time, \( {\varphi }_{y... | Yes |
Proposition 1.4 (Equality in the Schwarz inequality) Two vectors \( x \) and \( y \) in a scalar product space satisfy \( \left| \left( {x \mid y}\right) \right| = \parallel x\parallel \parallel y\parallel \) if and only if they are linearly dependent. | Proof. The \ | No |
Theorem 2.1 Let \( C \) be a nonempty, closed, convex subset of \( E \). For every point \( x \) of \( E \), there exists a unique point \( y \) of \( C \) such that\n\n\[ \parallel x - y\parallel = d\left( {x, C}\right) . \]\n\nThis point, called the projection of \( x \) onto \( C \) and denoted by \( {P}_{C}\left( x... | Proof. Fix \( x \in E \). We first show the existence of the projection of \( x \) onto \( C \). By the definition of \( \delta = d\left( {x, C}\right) \), there exists a sequence \( \left( {y}_{n}\right) \) in \( C \) such that\n\n\[ {\begin{Vmatrix}x - {y}_{n}\end{Vmatrix}}^{2} \leq {\delta }^{2} + \frac{1}{n}\;\text... | Yes |
Proposition 2.2 Under the assumptions of Theorem 2.1,\n\n\\begin{Vmatrix}{{P}_{C}\\left( {x}_{1}\\right) - {P}_{C}\\left( {x}_{2}\\right) }\\end{Vmatrix} \\leq \\begin{Vmatrix}{{x}_{1} - {x}_{2}}\\end{Vmatrix}\\;\\text{ for all }{x}_{1},{x}_{2} \\in E. | Proof. Set \\( {y}_{1} = {P}_{C}\\left( {x}_{1}\\right) \\) and \\( {y}_{2} = {P}_{C}\\left( {x}_{2}\\right) \\) . First,\n\n\\begin{align*}\n\\operatorname{Re}\\left( {{x}_{1} - {x}_{2} \\mid {y}_{1} - {y}_{2}}\\right) &= \\operatorname{Re}\\left( {{x}_{1} - {y}_{2} \\mid {y}_{1} - {y}_{2}}\\right) + \\operatorname{Re... | Yes |
Proposition 2.3 Let \( F \) be a closed vector subspace of \( E \) . Then \( {P}_{F} \) is a linear operator from \( E \) onto \( F \) . If \( x \in E \), the image \( {P}_{F}\left( x\right) \) is the unique element \( y \in E \) such that\n\n\[ y \in F\\text{ and }x - y \in {F}^{ \\bot }.\] | Proof. Condition \( \\left( *\\right) \) of Theorem 2.1 becomes\n\n\[ y \\in F\\;\\text{and}\\;\\operatorname{Re}\\left( {x - y \\mid z - y}\\right) \\leq 0\\;\\text{for all}z \\in F.\\]\n\nNow, if \( y \\in F \) and \( \\lambda \\in {\\mathbb{C}}^{ * } \), the map \( {z}^{\\prime } \\mapsto z = y + \\bar{\\lambda }{z}... | Yes |
Corollary 2.4 For every closed vector subspace \( F \) of \( E \), we have\n\n\[ E = F \oplus {F}^{ \bot } \]\n\nand the projection operator on \( F \) associated with this direct sum is \( {P}_{F} \) . | Proof. For \( x \in E \), we can write \( x = {P}_{F}\left( x\right) + \left( {x - {P}_{F}\left( x\right) }\right) \) and, by Proposition \( {2.3},{P}_{F}\left( x\right) \in F \) and \( x - {P}_{F}\left( x\right) \in {F}^{ \bot } \) . On the other hand, if \( x \in F \cap {F}^{ \bot } \) , then \( \left( {x \mid x}\rig... | Yes |
Corollary 2.5 For every vector subspace \( F \) of \( E \) , \[ E = \bar{F} \oplus {F}^{ \bot }. \] In particular, \( F \) is dense in \( E \) if and only if \( {F}^{ \bot } = \{ 0\} \) . | Proof. Just recall that \( {F}^{ \bot } = {\bar{F}}^{ \bot } \). | No |
Proposition 2.6 Let \( \mu \) be a positive Radon measure on a locally compact, separable metric space \( X \) . Then \( {C}_{c}\left( X\right) \) is dense in \( {L}^{2}\left( \mu \right) \) . | Proof. We write \( F = {C}_{c}\left( X\right) \) . If \( f \) is an element of \( {F}^{ \bot } \), then \( \int \varphi \bar{f}{d\mu } = 0 \) for all \( \varphi \in {C}_{c}\left( X\right) \) . Thus, for all \( \varphi \in {C}_{c}^{\mathbb{R}}\left( X\right) \) ,\n\n\[ \n\int \varphi {\left( \operatorname{Re}f\right) }^... | Yes |
Corollary 2.7 If \( E \) is a Hilbert space and \( F \) is a vector subspace of \( E \) , then \( \bar{F} = {F}^{ \bot \bot } \) . | Proof. Clearly \( F \subset {F}^{ \bot \bot } \) . Therefore, since \( {F}^{ \bot \bot } \) is closed, \( \bar{F} \subset {F}^{ \bot \bot } \) . On the other hand, we have \( E = \bar{F} \oplus {F}^{ \bot } \) and \( E = {F}^{ \bot \bot } \oplus {F}^{ \bot } \) . The result follows immediately. | Yes |
Theorem 3.1 (Riesz) The map from \( E \) to \( {E}^{\prime } \) defined by \( y \mapsto {\varphi }_{y} = \left( {\cdot \mid y}\right) \) is a surjective isometry. In other words, given any continuous linear form \( \varphi \) on \( E \), there exists a unique \( y \in E \) such that\n\n\[ \varphi \left( x\right) = \lef... | Proof. That this map is an isometry was seen in Corollary 1.3. We now show it is surjective. Take \( \varphi \in {E}^{\prime } \) such that \( \varphi \neq 0 \) . We know from Corollary 2.4 that \( E = \ker \varphi \oplus {\left( \ker \varphi \right) }^{ \bot } \), since, \( \varphi \) being continuous, \( \ker \varphi... | Yes |
Proposition 3.2 Given \( T \in L\left( E\right) \), there exists a unique operator \( {T}^{ * } \in \) \( L\left( E\right) \) such that\n\n\[ \left( {{Tx} \mid y}\right) = \left( {x \mid {T}^{ * }y}\right) \;\text{ for all }x, y \in E. \]\n\nMoreover, \( \begin{Vmatrix}{T}^{ * }\end{Vmatrix} = \parallel T\parallel \) .... | Proof. Take \( y \in E \) . The map \( {\varphi }_{y} \circ T : x \mapsto \left( {{Tx} \mid y}\right) \) is an element of \( {E}^{\prime } \), so by Theorem 3.1 there exists a unique element of \( E \), which we denote by \( {T}^{ * }y \), such that\n\n\[ \left( {{Tx} \mid y}\right) = \left( {x \mid {T}^{ * }y}\right) ... | Yes |
Proposition 3.4 For every \( T \in L\left( E\right) \), we have \( \begin{Vmatrix}{T{T}^{ * }}\end{Vmatrix} = \begin{Vmatrix}{{T}^{ * }T}\end{Vmatrix} = \parallel T{\parallel }^{2} \) . | Proof. Certainly \( \begin{Vmatrix}{{T}^{ * }T}\end{Vmatrix} \leq \parallel T{\parallel }^{2} \) . On the other hand,\n\n\[ \parallel {Tx}{\parallel }^{2} = \left( {{Tx} \mid {Tx}}\right) = \left( {x \mid {T}^{ * }{Tx}}\right) \leq \parallel x{\parallel }^{2}\begin{Vmatrix}{{T}^{ * }T}\end{Vmatrix}, \]\n\nwhich shows t... | Yes |
Proposition 3.5 Assume \( E \neq \{ 0\} \) . For every selfadjoint operator \( T \in \) \( L\left( E\right) \) , \n\n\[ \n\parallel T\parallel = \sup \{ \left| \left( {{Tx} \mid x}\right) \right| : x \in E\text{ and }\parallel x\parallel = 1\} . \n\] | Proof. Let \( \gamma \) be the right-hand side of the equality. Clearly \( \gamma \leq \parallel T\parallel \) and, for all \( x \in E,\left| \left( {{Tx} \mid x}\right) \right| \leq \gamma \parallel x{\parallel }^{2} \) . Assume for example that \( \mathbb{K} = \mathbb{C} \), and take \( y, z \in E \) and \( \lambda \... | Yes |
Proposition 3.6 Let \( \left( {x}_{n}\right) \) be a sequence in \( E \) that converges weakly to \( x \) . Then\n\n\[ \mathop{\liminf }\limits_{{n \rightarrow + \infty }}\begin{Vmatrix}{x}_{n}\end{Vmatrix} \geq \parallel x\parallel \]\n\nMoreover, the following properties are equivalent:\n\n1. The sequence \( \left( {... | Proof. First,\n\n\[ \parallel x{\parallel }^{2} = \mathop{\lim }\limits_{{n \rightarrow + \infty }}\left| \left( {x \mid {x}_{n}}\right) \right| \leq \parallel x\parallel \mathop{\liminf }\limits_{{n \rightarrow + \infty }}\begin{Vmatrix}{x}_{n}\end{Vmatrix}, \]\n\nwhich proves the first statement. At the same time, \(... | Yes |
Theorem 3.7 Any bounded sequence in \( E \) has a weakly convergent subsequence. | Proof. Suppose first that \( E \) is separable. Let \( \left( {x}_{n}\right) \) be a bounded sequence in \( E \) . In the notation of Theorem 3.1, the Banach–Alaoglu Theorem (page 19) applied to the sequence \( \left( {\varphi }_{{x}_{n}}\right) \) guarantees the existence of a subsequence \( \left( {x}_{{n}_{k}}\right... | Yes |
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