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Proposition 3.8 Let \( \\left( {x}_{n}\\right) \) be a sequence in \( E \) converging weakly to \( x \) . Then, for all \( T \\in L\\left( E\\right) \), the sequence \( \\left( {T{x}_{n}}\\right) \) converges weakly to \( {Tx} \) . | Proof. For every \( y \\in E \) ,\n\n\[ \n\\mathop{\\lim }\\limits_{{n \\rightarrow + \\infty }}\\left( {T{x}_{n} \\mid y}\\right) = \\mathop{\\lim }\\limits_{{n \\rightarrow + \\infty }}\\left( {{x}_{n} \\mid {T}^{ * }y}\\right) = \\left( {x \\mid {T}^{ * }y}\\right) = \\left( {{Tx} \\mid y}\\right) .\n\] | Yes |
Proposition 4.1 An orthogonal family that does not include the zero vector is free. | Proof. Let \( J \) be a finite subset of \( I \) and let \( {\left( {\lambda }_{j}\right) }_{j \in J} \) be elements of \( \mathbb{K} \) such that \( \mathop{\sum }\limits_{{j \in J}}{\lambda }_{j}{X}_{j} = 0 \) . Then\n\n\[ \n{\begin{Vmatrix}\mathop{\sum }\limits_{{j \in J}}{\lambda }_{j}{X}_{j}\end{Vmatrix}}^{2} = \m... | Yes |
Proposition 4.2 Let \( {\left\{ {e}_{j}\right\} }_{j \in J} \) be a finite orthonormal family in \( E \), spanning the vector subspace \( F \) . For every \( x \in E \), the orthogonal projection \( {P}_{F}\left( x\right) \) of \( x \) onto \( F \) is given by\n\n\[ \n{P}_{F}\left( x\right) = \mathop{\sum }\limits_{{j ... | Proof. To prove the first statement, it is enough to show that the vector \( y = \mathop{\sum }\limits_{{j \in J}}\left( {x \mid {e}_{j}\right) {e}_{j} \) satisfies the conditions characterizing \( {P}_{F}\left( x\right) \) (see Proposition 2.3 and the remark on page 107). Now, it is clear that \( y \in F \) and that \... | Yes |
Theorem 4.5 Let \( {\left( {e}_{i}\right) }_{i \in I} \) be a Hilbert basis of \( E \) . For any element \( x \) of \( E \) , \[ x = \mathop{\sum }\limits_{{i \in I}}\left( {x \mid {e}_{i}}\right) {e}_{i} \] | Proof. By Proposition 4.2, we know that, for any finite subset \( J \) of \( I \) , \[ {\begin{Vmatrix}x - \mathop{\sum }\limits_{{j \in J}}\left( x \mid {e}_{j}\right) {e}_{j}\end{Vmatrix}}^{2} = \parallel x{\parallel }^{2} - \mathop{\sum }\limits_{{j \in J}}{\left| \left( x \mid {e}_{j}\right) \right| }^{2}. \] Now j... | No |
Proposition 4.6 (Schmidt orthonormalization process) Suppose that \( N \in \{ 1,2,3,\ldots \} \cup \{ + \infty \} \) and let \( {\left( {f}_{n}\right) }_{0 \leq n < N} \) be a free family in \( E \) . There exists an orthonormal family \( {\left( {e}_{n}\right) }_{0 \leq n < N} \) of \( E \) such that, for each nonnega... | Such a family can be constructed by setting\n\n\[ \n{e}_{0} = \frac{1}{\begin{Vmatrix}{f}_{0}\end{Vmatrix}}{f}_{0} \n\] \n\nand, for \( 0 \leq n < N - 1 \) , \n\n\[ \n{x}_{n + 1} = {f}_{n + 1} - {P}_{n}{f}_{n + 1}\;\text{ and }\;{e}_{n + 1} = \frac{1}{\begin{Vmatrix}{x}_{n + 1}\end{Vmatrix}}{x}_{n + 1}, \n\] \n\nwhere ... | Yes |
Corollary 4.7 A scalar product space is separable if and only if it has a countable Hilbert basis. | Proof. According to Proposition 2.6 on page 10, the condition is sufficient. By the same proposition, separability implies the existence of a free and fundamental family \( {\left( {f}_{n}\right) }_{n \in \mathbb{N}} \) . Applying the Schmidt orthonormalization process to the family \( \left( {f}_{n}\right) \) we obtai... | Yes |
Proposition 1.4 If \( f \in {L}^{1} \cap {L}^{\infty } \), then \( f \in {L}^{p} \) for every \( p \in \left( {1,\infty }\right) \), and\n\n\[ \parallel f{\parallel }_{p} \leq \parallel f{\parallel }_{1}^{1/p}\parallel f{\parallel }_{\infty }^{1 - 1/p}. \]\n\nIn addition, if \( 1 \leq p < \infty ,{L}^{1} \cap {L}^{\inf... | Proof. If \( f \in {L}^{\infty } \) and \( 1 < p < \infty \), we clearly have \( {\left| f\right| }^{p} \leq \left| f\right| \parallel f{\parallel }_{\infty }^{p - 1} \) \( m \) -almost everywhere, which proves the first assertion of the proposition.\n\nNow suppose that \( 1 \leq p < \infty \) and that \( f \in {L}^{p}... | Yes |
Lemma 1.5 For each nonnegative real a, define a map \( {\Pi }_{a} : \mathbb{K} \rightarrow \mathbb{K} \) by setting \( {\Pi }_{0}\left( x\right) = 0 \) and\n\n\[ \n{\Pi }_{a}\left( x\right) = \frac{ax}{\max \left( {a,\left| x\right| }\right) }\;\text{ if }a > 0.\n\]\n\nThen, for every \( x \in \mathbb{K} \), we have \(... | Proof. It is clear that \( {\Pi }_{a} \) is exactly the projection map from the canonical euclidean space \( \mathbb{R} \) (or the canonical hermitian space \( \mathbb{C} \), as the case may be) onto \( \bar{B}\left( {0, a}\right) \) . The claims made are then obvious; the last of them can be seen as a particular case ... | No |
Theorem 1.6 If \( m \) is a Radon measure, the space \( {C}_{c}\left( X\right) \) is dense in \( {L}^{p}\left( m\right) \) for \( 1 \leq p < + \infty \) . | Proof. The case \( p = 1 \) was proved in Chapter 2, Proposition 3.5 on page 70. Suppose \( 1 < p < \infty \) . By Proposition 1.4, it suffices to approximate \( f \in \) \( {L}^{1} \cap {L}^{\infty } \) in the sense of the \( \parallel \cdot {\parallel }_{p} \) norm. Thus, fix \( f \in {L}^{1} \cap {L}^{\infty } \) an... | Yes |
Corollary 1.7 If \( m \) is a Radon measure, the space \( {L}^{p}\left( m\right) \) is separable for \( 1 \leq p < \infty \) . | Proof. Let \( \left( {K}_{n}\right) \) be a sequence of compact sets exhausting \( X \) . Since\n\n\[ \n{C}_{c}\left( X\right) = \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{C}_{{K}_{n}}\left( X\right) \n\]\n\nit suffices, by the preceding theorem, to show that each \( {C}_{{K}_{n}}\left( X\right) \) is separable with... | Yes |
Lemma 2.3 If \( T \in {\left( {L}^{p}\right) }^{\prime } \), there exists a measurable function \( g \) such that, for all \( f \in {L}^{p} \), \[ {fg} \in {L}^{1}\;\text{ and }\;{Tf} = \int {fgdm} \] | Proof. For \( f \in {L}_{\mathbb{R}}^{p} \), set \( {T}_{1}f = \operatorname{Re}\left( {Tf}\right) \) and \( {T}_{2}f = \operatorname{Im}\left( {Tf}\right) \). Then \( {T}_{1} \) and \( {T}_{2} \) belong to \( {\left( {L}_{\mathbb{R}}^{p}\right) }^{\prime } \). If Lemma 2.3 is true in the real case, we can apply it to ... | Yes |
Lemma 2.4 With the notation of Lemma 2.3, we have \( g \in {L}^{{p}^{\prime }} \) and \( \parallel g{\parallel }_{{p}^{\prime }} \leq \parallel T{\parallel }_{p}^{\prime } \), where \( \parallel \cdot {\parallel }_{p}^{\prime } \) is the norm in \( {\left( {L}^{p}\right) }^{\prime } \) . | Proof. Since the measure \( m \) is \( \sigma \) -finite, there exists an increasing sequence \( \left( {A}_{n}\right) \) of elements of \( \mathcal{F} \) of finite measure that cover \( X \) . 1. Case \( p = 1 \) . Suppose the conclusion of the lemma is false. Then the set \( \left\{ {\left| g\right| > \parallel T{\pa... | Yes |
If \( 1 \leq p \leq \infty \), the family \( {\left( {\tau }_{a}\right) }_{a \in {\mathbb{R}}^{d}} \) forms an abelian group of isometries of \( {L}^{p} \). | The first assertion follows immediately from the remarks preceding the theorem (in particular, from the translation invariance of \( \lambda \) ). | No |
Proposition 3.2 Let \( p,{p}^{\prime } \in \left\lbrack {1,\infty }\right\rbrack \) be conjugate exponents and suppose \( f \in {L}^{p} \) and \( g \in {L}^{{p}^{\prime }} \) . Then \( f * g \) is uniformly continuous and bounded, and\n\n\[ \parallel f * g{\parallel }_{\infty } \leq \parallel f{\parallel }_{p}\parallel... | Proof. The Hölder inequality yields\n\n\[ \left| {\left( {f * g}\right) \left( x\right) - \left( {f * g}\right) \left( {x}^{\prime }\right) }\right| \leq {\begin{Vmatrix}{\tau }_{x}\check{f} - {\tau }_{{x}^{\prime }}\check{f}\end{Vmatrix}}_{p}\parallel g{\parallel }_{{p}^{\prime }}\;\text{ for all }x,{x}^{\prime } \in ... | Yes |
Proposition 3.3 If \( f \) and \( g \) are convolvable equivalence classes of functions, | \[ \operatorname{Supp}\left( {f * g}\right) \subset \overline{\operatorname{Supp}f + \operatorname{Supp}g}. \] In particular, if \( f \) or \( g \) has compact support, we have \[ \operatorname{Supp}\left( {f * g}\right) \subset \operatorname{Supp}f + \operatorname{Supp}g \] (since, if \( F \) is closed and \( K \) is ... | No |
Proposition 3.5 Let \( p, q, r \in \left\lbrack {1, + \infty }\right\rbrack \) be such that \( 1/p + 1/q + 1/r \geq 2 \) . If \( f \in {L}^{p}, g \in {L}^{q} \), and \( h \in {L}^{r} \), then \( f * \left( {g * h}\right) \) and \( \left( {f * g}\right) * h \) are well defined and belong to \( {L}^{s} \), where \( s \) ... | Proof. That \( f * \left( {g * h}\right) \) and \( \left( {f * g}\right) * h \) are well defined and belong to \( {L}^{s} \) follows from Theorem 3.4. Next, \[ \left( {f * \left( {g * h}\right) }\right) \left( x\right) = \iint f\left( {x - y}\right) g\left( {y - z}\right) h\left( z\right) {dydz} \] \[ = \iint f\left( {... | No |
Corollary 3.6 The operations + and \( * \) make \( {L}^{1} \) into a commutative ring. | Proof. The convolution product is commutative and, by Theorem 3.4, \( {L}^{1} \) is closed under it. Proposition 3.5 says it is also associative. The rest is obvious. | No |
Proposition 3.7 Suppose \( p \in \lbrack 1,\infty ) \) and let \( {\left( {\varphi }_{n}\right) }_{n \in \mathbb{N}} \) be a Dirac sequence. If \( f \in {L}^{p} \), then\n\n\[ f * {\varphi }_{n} \in {L}^{p}\;\text{ and }\;{\begin{Vmatrix}f * {\varphi }_{n}\end{Vmatrix}}_{p} \leq \parallel f{\parallel }_{p}\;\text{ for ... | Proof. That \( f * {\varphi }_{n} \in {L}^{p} \) and \( {\begin{Vmatrix}f * {\varphi }_{n}\end{Vmatrix}}_{p} \leq \parallel f{\parallel }_{p} \) follows from Theorem 3.4. Further, for almost every \( x \) ,\n\n\[ \left| {f\left( x\right) - \left( {f * {\varphi }_{n}}\right) \left( x\right) }\right| \leq \int \left| {f\... | Yes |
Proposition 1.2 Suppose \( T \in L\left( E\right) \) . The limit \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{T}^{n}\end{Vmatrix}}^{1/n} \) exists and\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{T}^{n}\end{Vmatrix}}^{1/n} = \mathop{\inf }\limits_{{n \in {\mathbb{N}}^{ * }... | Proof\n\n1. Set \( a = \mathop{\inf }\limits_{{n \in {\mathbb{N}}^{ * }}}{\begin{Vmatrix}{T}^{n}\end{Vmatrix}}^{1/n} \) . Certainly we have\n\n\[ a \leq \mathop{\liminf }\limits_{{n \rightarrow + \infty }}{\begin{Vmatrix}{T}^{n}\end{Vmatrix}}^{1/n} \]\n\nTake \( \varepsilon > 0 \) and let \( {n}_{0} \in {\mathbb{N}}^{ ... | Yes |
Proposition 1.3 Suppose \( T \in L\left( E\right) \) . For all \( \lambda ,\mu \in \rho \left( T\right) \), we have\n\n\[ R\left( {\lambda, T}\right) - R\left( {\mu, T}\right) = \left( {\mu - \lambda }\right) R\left( {\lambda, T}\right) R\left( {\mu, T}\right) = \left( {\mu - \lambda }\right) R\left( {\mu, T}\right) R\... | Proof. First,\n\n\[ R\left( {\lambda, T}\right) - R\left( {\mu, T}\right) = R\left( {\lambda, T}\right) \left( {\left( {{\mu I} - T}\right) - \left( {{\lambda I} - T}\right) }\right) R\left( {\mu, T}\right) \]\n\n\[ = \left( {\mu - \lambda }\right) R\left( {\lambda, T}\right) R\left( {\mu, T}\right) \]\n\nwhich proves ... | Yes |
Theorem 1.5 (spectral image) If \( T \in L\left( E\right) \) and \( P \in \mathbb{K}\left\lbrack X\right\rbrack \), we have\n\n\[ P\left( {\sigma \left( T\right) }\right) \subset \sigma \left( {P\left( T\right) }\right) \]\n\nwith equality if \( \mathbb{K} = \mathbb{C} \) . | Proof\n\n1. Take \( \lambda \in \mathbb{K} \) . Since \( \lambda \) is a root of the polynomial \( P - P\left( \lambda \right) \), there exists a polynomial \( {Q}_{\lambda } \in \mathbb{K}\left\lbrack X\right\rbrack \) such that \( P - P\left( \lambda \right) = \left( {X - \lambda }\right) {Q}_{\lambda } \) . Then\n\n... | Yes |
Proposition 2.1 Suppose \( T \in L\left( E\right) \) . Then:\n\ni. \( \ker T = {\left( \operatorname{im}{T}^{ * }\right) }^{ \bot } \) .\n\nii. \( \overline{\operatorname{im}T} = {\left( \ker {T}^{ * }\right) }^{ \bot } \) .\n\niii. \( T \) is invertible if and only if \( {T}^{ * } \) is, and in this case\n\n\[{\left( ... | Proof. For \( x \in E \), we have \( x \in \ker T \) if and only if\n\n\[ \left( {{Tx} \mid y}\right) = \left( {x \mid {T}^{ * }y}\right) = 0\;\text{ for all }y \in E,\]\n\nwhich proves the first assertion. The second is a consequence of the first, in view of Corollary 2.7 on page 108 and of the equality \( {T}^{* * } ... | Yes |
Proposition 2.3 The spectral radius and the norm of a hermitian operator on \( E \) coincide. | Proof. If \( T \) is hermitian, Proposition 3.4 on page 113 says that \( \begin{Vmatrix}{T}^{2}\end{Vmatrix} = \) \( \parallel T{\parallel }^{2} \) . Iterating this property, which we can do because the square of a hermitian operator is hermitian, we obtain\n\n\[\n\begin{Vmatrix}{T}^{{2}^{n}}\end{Vmatrix} = \parallel T... | Yes |
Proposition 2.5 Every hermitian operator \( T \) on \( E \) has the following properties:\n\ni. The eigenvalues of \( T \) are real.\n\nii. For every \( \lambda \in \mathbb{C} \), we have \( \overline{\operatorname{im}\left( {{\lambda I} - T}\right) } = {\left( \ker \left( \bar{\lambda }I - T\right) \right) }^{ \bot } ... | Proof. Suppose that \( \lambda \) is an eigenvalue of \( T \), and let \( x \in E \) be an associated nonzero eigenvector, so that \( {Tx} = {\lambda x} \) and \( x \neq 0 \) . Then\n\n\[ \lambda \parallel x{\parallel }^{2} = \left( {{\lambda x} \mid x}\right) = \left( {{Tx} \mid x}\right) .\n\]\n\nSince the operator \... | Yes |
Proposition 2.8 For every \( P \in \mathbb{C}\left\lbrack X\right\rbrack \), we have \( {\left( P\left( T\right) \right) }^{ * } = \bar{P}\left( T\right) \) and \[ \parallel P\left( T\right) \parallel = \mathop{\max }\limits_{{t \in \sigma \left( T\right) }}\left| {P\left( t\right) }\right| \] | Proof. The first assertion is an immediate consequence of the fact that \( T \) is hermitian (see Proposition 3.3 on page 112). Next, for \( P \in \mathbb{C}\left\lbrack X\right\rbrack \) , Proposition 3.4 on page 113 gives \[ \parallel P\left( T\right) \parallel = {\begin{Vmatrix}P\left( T\right) P{\left( T\right) }^{... | Yes |
Corollary 2.10 Let \( f \) be a continuous function from \( \sigma \left( T\right) \) to \( \mathbb{C} \) . The operator \( f\left( T\right) \) is hermitian if and only if \( f \) is real-valued. It is positive hermitian if and only if \( f \geq 0 \) . | Proof. The first assertion follows from part ii of Theorem 2.9. The second follows from part iii of the same theorem and from Corollary 2.7. | No |
Proposition 1.1 \( \mathcal{K}\left( {E, F}\right) \) is a vector subspace of \( L\left( {E, F}\right) \) . | Proof. Consider compact operators \( T \) and \( S \) from \( E \) to \( F \) and elements \( \lambda ,\mu \in \mathbb{K} \) . Then\n\n\[ \left( {{\lambda T} + {\mu S}}\right) \left( {\bar{B}\left( E\right) }\right) \subset \lambda \overline{T\left( {\bar{B}\left( E\right) }\right) } + \mu \overline{S\left( {\bar{B}\le... | Yes |
Proposition 1.2 Let \( R \) be a compact operator from \( E \) to \( F \) . If \( {E}_{1} \) and \( {F}_{1} \) are normed spaces and if \( T \in L\left( {{E}_{1}, E}\right) \) and \( S \in L\left( {F,{F}_{1}}\right) \) are arbitrary, the composition SRT is a compact operator from \( {E}_{1} \) to \( {F}_{1} \) . | Proof. Indeed,\n\n\[ \n{SRT}\left( {\bar{B}\left( {E}_{1}\right) }\right) \subset \parallel T\parallel S\left( \overline{R\left( {\bar{B}\left( E\right) }\right) }\right) .\n\]\n\nSince a continuous image of a compact set is compact, the result follows. | Yes |
Proposition 1.4 If \( F \) is complete, the limit in \( L\left( {E, F}\right) \) of every convergent sequence of compact operators from \( E \) to \( F \) is a compact operator. | Proof. Let \( {\left( {T}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence of compact operators from \( E \) to \( F \) that converges to \( T \) in \( L\left( {E, F}\right) \) . By Theorem 3.3 on page 14, it suffices to show that \( T\left( {\bar{B}\left( E\right) }\right) \) is precompact. Choose \( \varepsilon > 0 \... | Yes |
Proposition 1.4 If \( F \) is complete, the limit in \( L\left( {E, F}\right) \) of every convergent sequence of compact operators from \( E \) to \( F \) is a compact operator. | Proof. Let \( {\left( {T}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence of compact operators from \( E \) to \( F \) that converges to \( T \) in \( L\left( {E, F}\right) \) . By Theorem 3.3 on page 14, it suffices to show that \( T\left( {\bar{B}\left( E\right) }\right) \) is precompact. Choose \( \varepsilon > 0 \... | Yes |
Corollary 1.5 If \( F \) is complete, every limit in \( L\left( {E, F}\right) \) of finite-rank operators is a compact operator. | This provides a frequently useful criterion for proving that an operator is compact. | No |
Lemma 1.7 If \( F \) is a proper closed subspace of a normed vector space \( G \), there exists \( u \in G \) such that \( \parallel u\parallel = 1 \) and \( d\left( {u, F}\right) \geq \frac{1}{2} \) . | Proof. Take \( v \in G \smallsetminus F \) and set \( \delta = d\left( {v, F}\right) > 0 \) . Certainly there exists \( w \in F \) such that \( \parallel v - w\parallel < {2\delta } \) . Then the point \( u = \parallel v - w{\parallel }^{-1}\left( {v - w}\right) \) works: if \( z \in F \), we have\n\n\[ \parallel u - z... | Yes |
Lemma 2.1 Let \( S \) be a compact selfadjoint operator on a scalar product space \( F \) not equal to \( \{ 0\} \) . Then \( S \) has at least one eigenvalue and\n\n\[ \max \{ \left| \lambda \right| : \lambda \in \operatorname{ev}\left( S\right) \} = \parallel S\parallel \] | Proof. Clearly, if \( \lambda \) is an eigenvalue of \( S \), then \( \left| \lambda \right| \leq \parallel S\parallel \) . On the other hand, we know from the remark following Theorem 2.6 on page 203 that there exists a spectral value \( \lambda \) of \( S \) such that \( \left| \lambda \right| = \mathop{\sup }\limits... | Yes |
Theorem 2.2 Let \( \Lambda \) be the set of eigenvalues of \( T \) . Write \( {\Lambda }^{ * } = \Lambda \smallsetminus \{ 0\} \) and, for each eigenvalue \( \lambda \), let \( {E}_{\lambda } \) be the eigenspace of \( T \) associated with \( \lambda \) .\n\n- \( \Lambda \) is a countable, infinite, bounded subset of \... | ## Proof\n\n1. That all eigenvalues are real and that eigenspaces associated with distinct eigenvalues are orthogonal comes from parts i and iii of Proposition 2.5 on page 203, whose proof did not use the completeness of \( E \) . That eigenspaces associated with nonzero eigenvalues are finite-dimensional comes from Th... | Yes |
Corollary 2.3 In the notation of Theorem 2.2,\n\n\\[ \n\\overline{\\operatorname{im}T} = \\overline{{\\bigoplus }_{\\lambda \\in {\\Lambda }^{ * }}{E}_{\\lambda }} \n\\] | Proof. We know that \\( {Tx} = \\mathop{\\sum }\\limits_{{\\lambda \\in {\\Lambda }^{ * }}}\\lambda {P}_{\\lambda }x \\) for every \\( x \\in E \\) . It follows that\n\n\\[ \n\\operatorname{im}T \\subset \\overline{{\\bigoplus }_{\\lambda \\in {\\Lambda }^{ * }}{E}_{\\lambda }},\\;\\text{ and hence }\\;\\overline{\\ope... | Yes |
Corollary 2.6 Suppose that \( E \) is complete. Let \( {P}_{0} \) be the operator of orthogonal projection onto \( {E}_{0} = \ker T \) . Then\n\n\[ x = \mathop{\sum }\limits_{{\lambda \in \Lambda }}{P}_{\lambda }x\;\text{ for all }x \in E \]\n\nand\n\n\[ E = \overline{{\bigoplus }_{\lambda \in \Lambda }{E}_{\lambda }} ... | Proof. Since \( T \) is selfadjoint, we have \( {E}_{0} = \ker T = {\overline{\operatorname{im}T}}^{ \bot } \) . Therefore, if \( E \) is complete, \( E = {E}_{0} \oplus \overline{\operatorname{im}T} \) by Corollary 2.4 on page 107 .\n\nIf, moreover, \( E \) is separable, so is \( \ker T \) . Thus \( \ker T \) has a co... | Yes |
Corollary 2.6 Suppose that \( E \) is complete. Let \( {P}_{0} \) be the operator of orthogonal projection onto \( {E}_{0} = \ker T \) . Then\n\n\[ x = \mathop{\sum }\limits_{{\lambda \in \Lambda }}{P}_{\lambda }x\;\text{ for all }x \in E \]\n\nand\n\n\[ E = \overline{{\bigoplus }_{\lambda \in \Lambda }{E}_{\lambda }} ... | Proof. Since \( T \) is selfadjoint, we have \( {E}_{0} = \ker T = {\overline{\operatorname{im}T}}^{ \bot } \) . Therefore, if \( E \) is complete, \( E = {E}_{0} \oplus \overline{\operatorname{im}T} \) by Corollary 2.4 on page 107 . | No |
Proposition 2.8 With the notation and hypotheses above,\n\n\[ \iint {\left| K\left( x, y\right) \right| }^{2}{dm}\left( x\right) {dm}\left( y\right) = \mathop{\sum }\limits_{{n = 0}}^{{+\infty }}{\mu }_{n}^{2} = \mathop{\sum }\limits_{\substack{{\lambda \in \mathrm{{ev}}\left( T\right) } \\ {\lambda \neq 0} }}{d}_{\lam... | Proof. Take \( u \in \ker T \) . For almost every \( y \), the function \( {K}_{y} : x \mapsto K\left( {x, y}\right) \) lies in \( E \) and\n\n\[ \left( {{K}_{y} \mid u}\right) = \int K\left( {x, y}\right) \bar{u}\left( x\right) {dm}\left( x\right) = \overline{Tu}\left( y\right) = 0. \]\n\nThe second of these equalitie... | Yes |
Proposition 2.9 We have\n\n\[ K\left( {x, y}\right) = \mathop{\sum }\limits_{{n = 0}}^{{+\infty }}{\mu }_{n}{f}_{n}\left( x\right) \overline{{f}_{n}}\left( y\right) \]\n\nthe series being convergent in \( {L}^{2}\left( {m \times m}\right) \) . | Proof. Set \( {K}_{N}\left( {x, y}\right) = \mathop{\sum }\limits_{{n = 0}}^{N}{\mu }_{n}{f}_{n}\left( x\right) \overline{{f}_{n}}\left( y\right) \) . By equality (*) above, for almost every \( y \), we have\n\n\[ {K}_{y} = \mathop{\sum }\limits_{{n = 0}}^{{+\infty }}{\mu }_{n}\overline{{f}_{n}}\left( y\right) {f}_{n} ... | Yes |
Proposition 2.10 Suppose that \( \Phi : x \mapsto \int {\left| K\left( x, y\right) \right| }^{2}{dm}\left( y\right) \) belongs to \( {L}^{\infty }\left( m\right) \) . Then, for every \( n \in \mathbb{N} \), we have \( {f}_{n} \in {L}^{\infty }\left( m\right) \) and \[ f = \mathop{\sum }\limits_{{n = 0}}^{{+\infty }}\le... | Proof. For every \( n \in \mathbb{N} \) we have \[ {f}_{n}\left( x\right) = \frac{1}{{\mu }_{n}}\int K\left( {x, y}\right) {f}_{n}\left( y\right) {dm}\left( y\right) . \] Therefore \( {f}_{n} \in {L}^{\infty }\left( m\right) \) and \( {\begin{Vmatrix}{f}_{n}\end{Vmatrix}}_{\infty } \leq {\mu }_{n}^{-1}\sqrt{L} \), wher... | Yes |
Proposition 1.1 (Leibniz’s formula) Suppose \( f, g \in {\mathcal{E}}^{m}\left( \Omega \right) \) . For each multiindex \( p \) such that \( \left| p\right| \leq m \) , | \[ {D}^{p}\left( {fg}\right) = \mathop{\sum }\limits_{{q \leq p}}\left( \begin{array}{l} p \\ q \end{array}\right) {D}^{p - q}f{D}^{q}g. \] | Yes |
Proposition 1.2 Assume \( \varphi \in {\mathcal{D}}^{m} \), for some \( m \in \mathbb{N} \) . For every integer \( n \geq 1 \), the convolution \( \varphi * {\chi }_{n} \) belongs to \( \mathcal{D} \) and \[ \mathop{\lim }\limits_{{n \rightarrow + \infty }}\varphi * {\chi }_{n} = \varphi \;\text{ in }{\mathcal{D}}^{m} ... | Proof. Since the functions \( \varphi \) and \( {\chi }_{n} \) have compact support, so does \( \varphi * {\chi }_{n} \) . More precisely, \[ \operatorname{Supp}\left( {\varphi * {\chi }_{n}}\right) \subset \operatorname{Supp}\varphi + \operatorname{Supp}{\chi }_{n} \subset \operatorname{Supp}\varphi + \bar{B}\left( {0... | Yes |
Corollary 1.3 For every \( n \in \mathbb{N} \), the space \( \mathcal{D}\left( \Omega \right) \) is dense in \( {\mathcal{D}}^{m}\left( \Omega \right) \) . In particular, \( \mathcal{D}\left( \Omega \right) \) is dense in \( {C}_{c}\left( \Omega \right) \) . | Proof. If \( \varphi \in {\mathcal{D}}^{m}\left( \Omega \right) \), we can consider \( \varphi \) as an element of \( {\mathcal{D}}^{m} \) (by extending it with the value 0 on \( {\mathbb{R}}^{d} \smallsetminus \Omega \) ). Now\n\n\[ \operatorname{Supp}\left( {\varphi * {\chi }_{n}}\right) \subset \operatorname{Supp}\v... | Yes |
Proposition 1.4 If \( K \) is a compact subset of \( {\mathbb{R}}^{d} \) and \( {O}_{1},\ldots ,{O}_{n} \) are open sets in \( {\mathbb{R}}^{d} \) such that \( K \subset \mathop{\bigcup }\limits_{{j = 1}}^{n}{O}_{j} \), there exist functions \( {\varphi }_{1},\ldots ,{\varphi }_{n} \) in \( \mathcal{D} \) such that\n\n... | Proof. Set \( d = d\left( {K,{\mathbb{R}}^{d} \smallsetminus O}\right) \), with \( O = \mathop{\bigcup }\limits_{{j = 1}}^{n}{O}_{j} \) (the metric being the canonical euclidean metric in \( \left. {\mathbb{R}}^{d}\right) \) . Set \( {K}^{\prime } = \{ x : d\left( {x, K}\right) \leq d/2\} \) . The set \( {K}^{\prime } ... | Yes |
Proposition 1.5 The space \( \mathcal{D}\left( \Omega \right) \) is dense in \( \mathcal{E}\left( \Omega \right) \) and in \( {\mathcal{E}}^{m}\left( \Omega \right) \), for every \( m \in \mathbb{N} \) . | Proof. Let \( {\left( {K}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence of compact subsets of \( \Omega \) exhausting \( \Omega \) . By the previous proposition, there exists, for every integer \( n \in \mathbb{N} \), an element \( {\varphi }_{n} \in \mathcal{D}\left( \Omega \right) \) such that\n\n\[ 0 \leq {\varph... | Yes |
Proposition 2.1 Let \( T \) be a linear form on \( \mathcal{D}\left( \Omega \right) \). Then \( T \) is a distribution on \( \Omega \) if and only if, for every compact \( K \) in \( \Omega \), there exist \( m \in \mathbb{N} \) and \( C \geq 0 \) such that \[ \left| {T\left( \varphi \right) }\right| \leq C\parallel \v... | Proof. The \ | No |
Proposition 2.2 Two locally integrable functions on \( \Omega \) define the same distribution if and only if they coincide almost everywhere. | Proof. Take \( f \in {L}_{\text{loc }}^{1}\left( \Omega \right) \) such that \( \left\lbrack f\right\rbrack = 0 \) . Because \( \mathcal{D}\left( \Omega \right) \) is dense in \( {C}_{c}\left( \Omega \right) = {\mathcal{D}}^{0}\left( \Omega \right) \) (Corollary 1.3), we see that\n\n\[ \n{\int }_{\Omega }g\left( x\righ... | Yes |
Proposition 2.3 Every positive distribution has order 0. | Proof. Let \( T \) be a positive distribution on \( \Omega \) . Let \( K \) be a compact subset of \( \Omega \) and let \( \rho \in \mathcal{D}\left( \Omega \right) \) be such that \( 0 \leq \rho \leq 1 \) and \( \rho = 1 \) on \( K \) . For every \( \varphi \in {\mathcal{D}}_{K}\left( \Omega \right) \), we have \( \le... | Yes |
For every \( \varphi \in \mathcal{D}\left( \mathbb{R}\right) \), the limit\n\n\[ \langle T,\varphi \rangle = \mathop{\lim }\limits_{{\varepsilon \rightarrow {0}^{ + }}}\left( {{\int }_{\varepsilon }^{+\infty }\frac{\varphi \left( x\right) }{x}{dx} + \varphi \left( 0\right) \log \varepsilon }\right) \]\n\nexists. The li... | Proof. Take \( \varphi \in \mathcal{D}\left( \mathbb{R}\right) \) and \( A > 0 \) such that \( \operatorname{Supp}\varphi \subset \left\lbrack {-A, A}\right\rbrack \). Then\n\n\[ {\int }_{\varepsilon }^{+\infty }\frac{\varphi \left( x\right) }{x}{dx} = {\int }_{\varepsilon }^{A}\frac{\varphi \left( x\right) - \varphi \... | Yes |
For every \( \varphi \in \mathcal{D}\left( {\mathbb{R}}^{2}\right) \), the limit\n\n\[ \langle T,\varphi \rangle = \mathop{\lim }\limits_{{\varepsilon \rightarrow {0}^{ + }}}\left( {{\iint }_{\{ r \geq \varepsilon \} }{r}^{-4}{\varphi dxdy} - {\pi \varphi }\left( {0,0}\right) {\varepsilon }^{-2} + \frac{\pi }{2}{\Delta... | Summary of proof. Take \( \varphi \in \mathcal{D}\left( {\mathbb{R}}^{2}\right) \) and \( A > 0 \) such that \( \operatorname{Supp}\varphi \subset \) \( \bar{B}\left( {0, A}\right) \) . A quick calculation shows that\n\n\[ {\iint }_{\{ r \geq \varepsilon \} }\frac{\varphi }{{r}^{4}}\;{dx}\;{dy} \]\n\n\[ = {\iint }_{\{ ... | Yes |
Proposition 3.1 Let \( T \) be a distribution on \( \Omega \) and suppose \( m \in \mathbb{N} \) . A necessary and sufficient condition for \( T \) to have order at most \( m \) is that \( T \) can be extended to a continuous linear form on \( {\mathcal{D}}^{m}\left( \Omega \right) \) . The extension is then unique. | Proof. Suppose that \( T \) has order at most \( m \) . Property \( \left( *\right) \) on page 268 then implies that \( T \) is continuous (and even uniformly continuous) on the space \( \mathcal{D}\left( \Omega \right) \) regarded, topologically speaking, as a subspace of \( {\mathcal{D}}^{m}\left( \Omega \right) \) .... | Yes |
Proposition 3.4 Every distribution \( T \) with compact support in \( \Omega \) has finite order. More precisely, there exists an integer \( m \in \mathbb{N} \) and a constant \( C \geq 0 \) such that\n\n\[ \n\\left| {\\langle T,\\varphi \\rangle }\\right| \\leq C\\parallel \\varphi {\\parallel }^{\\left( m\\right) }\\... | Proof. Let \( K \) be the support of \( T \) and let \( {K}^{\\prime },{K}^{\\prime \\prime } \) be compact sets such that\n\n\[ \nK \\subset {\\mathring{K}}^{\\prime } \\subset {K}^{\\prime } \\subset {\\mathring{K}}^{\\prime \\prime } \\subset {K}^{\\prime \\prime } \\subset \\Omega .\n\]\n\nBy Proposition 2.1, there... | Yes |
Lemma 1.1 Suppose \( \alpha \in \mathcal{E}\left( \Omega \right) \) . The map \( \varphi \mapsto {\alpha \varphi } \) from \( \mathcal{D}\left( \Omega \right) \) to \( \mathcal{D}\left( \Omega \right) \) is continuous. Likewise, if \( \alpha \in {\mathcal{E}}^{m}\left( \Omega \right) \), with \( m \in \mathbb{N} \), th... | In other words, if \( {\left( {\varphi }_{n}\right) }_{n \in \mathbb{N}} \) is a sequence in \( \mathcal{D}\left( \Omega \right) \) or \( {\mathcal{D}}^{m}\left( \Omega \right) \) converging to 0 in \( \mathcal{D}\left( \Omega \right) \) or \( {\mathcal{D}}^{m}\left( \Omega \right) \), respectively, the same is true ab... | Yes |
Proposition 1.3 With the notation introduced in Definition 1.2, we have\n\n\[ \operatorname{Supp}\left( {\alpha T}\right) \subset \operatorname{Supp}\alpha \cap \operatorname{Supp}T \] | Proof. The second claim is obvious. To show the first, take \( \varphi \in \mathcal{D}\left( \Omega \right) \) . If \( \operatorname{Supp}\varphi \subset \Omega \smallsetminus \operatorname{Supp}\alpha \), then \( {\alpha \varphi } = 0 \), so \( \langle {\alpha T},\varphi \rangle = 0 \) . It follows that \( \Omega \sma... | Yes |
Proposition 1.4 For every \( S \in {\mathcal{D}}^{\prime }\left( \mathbb{R}\right) \), there exists \( T \in {\mathcal{D}}^{\prime }\left( \mathbb{R}\right) \) such that \( {xT} = S \) . If \( {T}_{0} \) is such that \( x{T}_{0} = S \), the set of solutions of the equation \( {xT} = S \) equals \( \left\{ {{T}_{0} + {C... | Proof. Take \( \chi \in \mathcal{D}\left( \mathbb{R}\right) \) such that \( \chi \left( 0\right) = 1 \) . To each \( \varphi \in \mathcal{D}\left( \mathbb{R}\right) \) we associate \( \widetilde{\varphi } \), defined by\n\n\[ \widetilde{\varphi }\left( x\right) = {\int }_{0}^{1}\left( {{\varphi }^{\prime }\left( {tx}\r... | Yes |
Proposition 1.5 Suppose \( T \in {\mathcal{D}}^{\prime }\left( \mathbb{R}\right) \) . Then \( {xT} = 1 \) if and only if there exists \( C \in \mathbb{C} \) such that \( T = \operatorname{pv}\left( {1/x}\right) + {C\delta } \) . | Proof. By Proposition 1.4, it suffices to show that \( x\mathrm{{pv}}\left( {1/x}\right) = 1 \) . To do this, take \( \varphi \in \mathcal{D}\left( \mathbb{R}\right) \) . By definition,\n\n\[ \langle x\operatorname{pv}\left( {1/x}\right) ,\varphi \rangle = \langle \operatorname{pv}\left( {1/x}\right) ,{x\varphi }\rangl... | Yes |
Proposition 2.1 Suppose \( m \in \mathbb{N} \) . For every \( T \in {\mathcal{D}}^{\prime m}\left( \Omega \right) \), we have \( {D}^{p}T \in {\mathcal{D}}^{\prime m + \left| p\right| }\left( \Omega \right) \) and \[ \left\langle {{D}^{p}T,\varphi }\right\rangle = {\left( -1\right) }^{\left| p\right| }\left\langle {T,{... | Proof. By Fubini’s Theorem, we can reduce to the case \( d = 1 \), to which we apply the classical theorem of integration by parts, taking into account that the support of \( \varphi \) is a compact subset of \( \Omega \), so that the \ | No |
Proposition 2.2 Let \( m \in \mathbb{N} \) and \( p \in {\mathbb{N}}^{d} \) satisfy \( \left| p\right| \leq m \) . If \( f \in {\mathcal{E}}^{m}\left( \Omega \right) \) , then\n\n\[ \n{D}^{p}\left( \left\lbrack f\right\rbrack \right) = \left\lbrack {{D}^{p}f}\right\rbrack \n\]\n\nIn this equality, the first \( {D}^{p} ... | The proposition is easily obtained by induction on \( \left| p\right| \) starting from the case \( \left| p\right| = 1 \), which is a consequence of the following lemma.\n\nLemma 2.3 (Integration by parts) If \( f \in {\mathcal{E}}^{1}\left( \Omega \right) \) and \( \varphi \in {\mathcal{D}}^{1}\left( \Omega \right) \)... | Yes |
Lemma 2.3 (Integration by parts) If \( f \in {\mathcal{E}}^{1}\left( \Omega \right) \) and \( \varphi \in {\mathcal{D}}^{1}\left( \Omega \right) \) , then, for every \( j \in \{ 1,\ldots, d\} \) , \[ {\int }_{\Omega }{D}_{j}{f\varphi dx} = - {\int }_{\Omega }f{D}_{j}{\varphi dx} \] | Proof. By Fubini’s Theorem, we can reduce to the case \( d = 1 \), to which we apply the classical theorem of integration by parts, taking into account that the support of \( \varphi \) is a compact subset of \( \Omega \), so that the \ | No |
Proposition 2.5 (Leibniz’s formula) Consider \( T \in {\mathcal{D}}^{\prime }\left( \Omega \right) ,\alpha \in \mathcal{E}\left( \Omega \right) \) , and \( p \in {\mathbb{N}}^{d}. \) Then\n\n\[ \n{D}^{p}\left( {\alpha T}\right) = \mathop{\sum }\limits_{{q \leq p}}\left( \begin{array}{l} p \\ q \end{array}\right) {D}^{p... | Proof. This is obvious if \( \left| p\right| = 0 \) . Consider the case \( \left| p\right| = 1 \) . If \( j \in \{ 1,\ldots, d\} \) , we have\n\n\[ \n\left\langle {{D}_{j}\left( {\alpha T}\right) ,\varphi }\right\rangle = - \left\langle {{\alpha T},{D}_{j}\varphi }\right\rangle = - \left\langle {T,\alpha {D}_{j}\varphi... | Yes |
Theorem 2.7 Let \( \Omega \) be a connected open subset of \( {\mathbb{R}}^{d} \) and suppose that \( T \) is a distribution on \( \Omega \) such that \( {D}_{j}T = 0 \) for every \( j \in \{ 1,\ldots, d\} \) . Then \( T = C \) for some \( C \in \mathbb{C} \) . | Proof. By the preceding theorem, there exists \( f \in {\mathcal{E}}^{1}\left( \Omega \right) \) such that \( T = \left\lbrack f\right\rbrack \) and \( {D}_{j}f = 0 \) in the ordinary sense, for all \( j \in \{ 1,\ldots, d\} \) . The result follows. | No |
Theorem 2.8 Suppose that \( \Omega \) is an open interval in \( \mathbb{R} \) and that \( \alpha \in \Omega \) . Let \( T \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and \( f \in {L}_{\mathrm{{loc}}}^{1}\left( \Omega \right) \) . The following properties are equivalent:\n\ni. \( {T}^{\prime } = \left\lbrack f\r... | Proof. Suppose \( \Omega = \left( {a, b}\right) \) . Take \( f \in {L}_{\text{loc }}^{1}\left( \left( {a, b}\right) \right) \) and let \( F\left( x\right) = {\int }_{\alpha }^{x}f\left( t\right) {dt} \) . Then, for every \( \varphi \in \mathcal{D}\left( \left( {a, b}\right) \right) \) ,\n\n\[ \left\langle {{\left\lbrac... | Yes |
Theorem 2.9 Suppose that \( \Omega \) is an open interval in \( \mathbb{R} \), and that \( T \in \) \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) . If there exists an increasing function \( \alpha \) on \( \Omega \) such that \( T = \left\lbrack \alpha \right\rbrack \) , then \( {T}^{\prime } = {d\alpha } \) and t... | Proof. Set \( \Omega = \left( {a, b}\right) \) . Let \( \alpha \) be an increasing function on \( \left( {a, b}\right) \) . Take \( \varphi \in \) \( \mathcal{D}\left( \Omega \right) \) and let \( c, d \) be such that \( a < c < d < b \) and the support of \( \varphi \) is contained is \( \left\lbrack {c, d}\right\rbra... | Yes |
Theorem 2.10 Suppose that \( \Omega \) is an open subset of \( \mathbb{R} \) and that \( f \) is a function on \( \Omega \) for which there exist points \( {x}_{1} < \cdots < {x}_{n} \) in \( \Omega \) satisfying these conditions:\n\n- \( f \) is of class \( {C}^{1} \) on \( \Omega \smallsetminus \left\{ {{x}_{1},\ldot... | Proof. Considering separately each of the connected components of \( \Omega \), we can assume that \( \Omega \) is an open interval \( \left( {a, b}\right) \) . Put \( {x}_{0} = a \) and \( {x}_{n + 1} = b \) . Then, if \( \varphi \in \mathcal{D}\left( \Omega \right) \), we have\n\n\[ \n\left\langle {{\left\lbrack f\ri... | Yes |
Theorem 2.12 Suppose that \( d \geq 2 \) and, if \( \left( {{x}_{2},\ldots ,{x}_{d}}\right) \in {\mathbb{R}}^{d - 1} \), write\n\n\[{\Omega }_{{x}_{2},\ldots ,{x}_{d}} = \left\{ {{x}_{1} \in \mathbb{R} : \left( {{x}_{1},{x}_{2},\ldots ,{x}_{d}}\right) \in \Omega }\right\} .\n\]\n\nLet \( f \in {L}_{\text{loc }}^{1}\lef... | Proof. Argue as in the proof of Theorem 2.10 and apply Fubini’s Theorem. | No |
Theorem 3.1 Let \( P\left( X\right) = \mathop{\sum }\limits_{{j = 0}}^{m}{a}_{j}{X}^{j} \), where \( m \in {\mathbb{N}}^{ * },{a}_{1},\ldots ,{a}_{m} \in \mathbb{C} \) , and \( {a}_{m} \neq 0 \) . Let \( \varphi \) be the solution on \( \mathbb{R} \) of the differential equation\n\n\[ \mathop{\sum }\limits_{{j = 0}}^{m... | Proof. As a particular case of Example 4 on page 301, we have\n\n\[ {\left\lbrack Y\varphi \right\rbrack }^{\left( m\right) } = \left\lbrack {Y{\varphi }^{\left( m\right) }}\right\rbrack + \delta \]\n\n\[ {\left\lbrack Y\varphi \right\rbrack }^{\left( k\right) } = \left\lbrack {Y{\varphi }^{\left( k\right) }}\right\rbr... | Yes |
Theorem 3.4 In \( {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{2}\right) \), \[ \frac{\partial }{\partial \bar{z}}\left( \frac{1}{\pi z}\right) = \delta \] | Proof. We follow a method analogous to the one used in the proof of Theorem 3.2. For \( \varepsilon > 0 \), put \[ {f}_{\varepsilon }\left( {x, y}\right) = \left\{ \begin{array}{ll} 1/z & \text{ if }\left| z\right| > \varepsilon \\ \bar{z}/{\varepsilon }^{2} & \text{ if }\left| z\right| \leq \varepsilon \end{array}\rig... | Yes |
Theorem 1.1 (Differentiation inside the brackets) Let \( m \in \mathbb{N} \) and \( r \in \mathbb{N} \) . If \( T \in {\mathcal{D}}^{\prime m}\left( \Omega \right) \) and \( \varphi \in {\mathcal{D}}^{m + r}\left( {\Omega \times {\Omega }^{\prime }}\right) \), the map on \( {\Omega }^{\prime } \) defined by\n\n\[ y \ma... | Proof. We carry out the proof in the case \( T \in {\mathcal{D}}^{\prime m}\left( \Omega \right) ,\varphi \in {\mathcal{D}}^{m + r}\left( {\Omega \times {\Omega }^{\prime }}\right) \) . The other case is very similar.\n\nCase \( r = 0 \) . Take \( T \in {\mathcal{D}}^{\prime m}\left( \Omega \right) \) and \( \varphi \i... | Yes |
Theorem 1.2 The vector space \( \mathcal{D}\left( \Omega \right) \otimes \mathcal{D}\left( {\Omega }^{\prime }\right) \) spanned by the functions\n\n\[ f \otimes g : \left( {x, y}\right) \mapsto f\left( x\right) g\left( y\right) \]\n\nwith \( f \in \mathcal{D}\left( \Omega \right) \) and \( g \in \mathcal{D}\left( {\Om... | Proof. We use a lemma that allows us to approximate the convolution by means of a \ | No |
Lemma 1.3 Suppose \( \varphi ,\psi \in \mathcal{D}\left( {\mathbb{R}}^{n}\right) \) . For \( \varepsilon > 0 \) and \( x \in {\mathbb{R}}^{n} \), set\n\n\[ \n{g}_{\varepsilon }\left( x\right) = {\varepsilon }^{n}\mathop{\sum }\limits_{{\nu \in {\mathbb{Z}}^{n}}}\varphi \left( {x - {\varepsilon \nu }}\right) \psi \left(... | Proof. The function \( {g}_{\varepsilon } \) is defined by a finite sum whose number of terms depends only on \( \varepsilon \) (since \( \psi \) has compact support). Since each of these terms is an element of \( \mathcal{D}\left( {\mathbb{R}}^{n}\right) \) and is supported within Supp \( \varphi + \operatorname{Supp}... | Yes |
Proposition 1.4 Suppose \( T \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and \( S \in {\mathcal{D}}^{\prime }\left( {\Omega }^{\prime }\right) \). There exists a unique distribution on \( \Omega \times {\Omega }^{\prime } \), denoted \( T \otimes S \) and called the tensor product of \( T \) and \( S \), such t... | Proof. Uniqueness follows immediately from Theorem 1.2. For existence, consider the linear map on \( \mathcal{D}\left( {\Omega \times {\Omega }^{\prime }}\right) \) defined by\n\n\[ \varphi \mapsto \left\langle {{S}_{y},\left\langle {{T}_{x},\varphi \left( {x, y}\right) }\right\rangle }\right\rangle \]\n\n(*) \n\nThis ... | Yes |
Proposition 1.5 Suppose \( T \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and \( S \in {\mathcal{D}}^{\prime }\left( {\Omega }^{\prime }\right) \) . Then:\n\ni. \( \operatorname{Supp}\left( {T \otimes S}\right) = \left( {\operatorname{Supp}T}\right) \times \left( {\operatorname{Supp}S}\right) \) .\n\nii. For any... | Proof. If \( \varphi \) is supported within \( \left( {\Omega \smallsetminus \operatorname{Supp}T}\right) \times {\Omega }^{\prime } \), the support of \( \varphi \left( {\cdot, y}\right) \) , for every \( y \in {\Omega }^{\prime } \), is contained in \( \Omega \smallsetminus \operatorname{Supp}T \) . Therefore \[ \lan... | Yes |
Proposition 2.2 If \( T, S \in {\mathcal{E}}^{\prime }\left( {\mathbb{R}}^{d}\right) \), then \( T * S \in {\mathcal{E}}^{\prime }\left( {\mathbb{R}}^{d}\right) \) and\n\n\[ \operatorname{Supp}\left( {T * S}\right) \subset \operatorname{Supp}T + \operatorname{Supp}S \] | Proof. Let \( \varphi \in \mathcal{D}\left( {\mathbb{R}}^{d}\right) \) . If \( x \notin \operatorname{Supp}\varphi - \operatorname{Supp}S \), then\n\n\[ \operatorname{Supp}\varphi \left( {x + \cdot }\right) \cap \operatorname{Supp}S = \left( {\operatorname{Supp}\varphi - x}\right) \cap \operatorname{Supp}S = \varnothin... | Yes |
Proposition 2.5 Let \( \Omega \) be open in \( {\mathbb{R}}^{d} \) . Let \( T \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and \( \varphi \in \mathcal{E}\left( \Omega \right) \) be such that \( \operatorname{Supp}T \cap \operatorname{Supp}\varphi \) is compact. Then, if \( \rho \in \mathcal{D}\left( \Omega \righ... | Proof. Take \( \rho \in \mathcal{D}\left( \Omega \right) \) such that \( \rho = 0 \) on an open set containing \( \operatorname{Supp}T \cap \) Supp \( \varphi \) . Then the support of \( \rho \) is contained in the complement of Supp \( T \cap \) Supp \( \varphi \), and therefore\n\n\[ \operatorname{Supp}{\rho \varphi ... | Yes |
Proposition 2.6 Let \( \left( {{T}_{1},\ldots ,{T}_{n}}\right) \) be a family of distributions on \( {\mathbb{R}}^{d} \) satisfying condition \( \left( \mathrm{C}\right) \) . 1. If \( \varphi \in \mathcal{D}\left( {\mathbb{R}}^{d}\right) \), we define a function \( \widehat{\varphi } \) on \( {\left( {\mathbb{R}}^{d}\r... | Proof. Let \( \Omega \) be a bounded open set in \( {\mathbb{R}}^{d} \) and \( \varphi \) an element of \( \mathcal{D}\left( \Omega \right) \) . We know that \( \operatorname{Supp}\left( {{T}_{1} \otimes \cdots \otimes {T}_{n}}\right) = \operatorname{Supp}{T}_{1} \times \cdots \times \operatorname{Supp}{T}_{n} \), so \... | Yes |
Proposition 2.8 (Continuity) Let \( {\left( {T}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence in \( {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{d}\right) \) , and let \( T, S \) belong to \( {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{d}\right) \) . Suppose that the sequence \( {\left( {T}_{n}\right) }_{n \in \mathb... | Proof. Take \( \varphi \in \mathcal{D}\left( {\mathbb{R}}^{d}\right) \) . As above, write \( \widehat{\varphi }\left( {x, y}\right) = \varphi \left( {x + y}\right) \) . Since the family \( \left( {F,\operatorname{Supp}S}\right) \) satisfies (C), the intersection \( \operatorname{Supp}\widehat{\varphi } \cap \left( {F \... | Yes |
Proposition 2.9 Suppose \( \\left( {T, S}\\right) \) satisfies property \( \\left( \\mathrm{C}\\right) \) . Then, for every \( \\varphi \\in \\mathcal{D}\\left( {\\mathbb{R}}^{d}\\right) \), the function \( \\widetilde{\\varphi } \) on \( {\\mathbb{R}}^{d} \) defined by\n\n\[ \n\\widetilde{\\varphi }\\left( x\\right) =... | Proof. Put \( K = \\{ \\left( {x, y}\\right) \\in \\operatorname{Supp}T \\times \\operatorname{Supp}S : x + y \\in \\operatorname{Supp}\\varphi \\} \) . Then the support of \( \\widetilde{\\varphi } \) is contained in Supp \( \\varphi \) -Supp \( S \) and \( \\left( {\\operatorname{Supp}\\varphi \\text{-Supp}S}\\right)... | Yes |
Corollary 2.10 Let \( f \) and \( g \) be elements of \( {L}_{\mathrm{{loc}}}^{1}\left( {\mathbb{R}}^{d}\right) \) whose supports satisfy condition (C). Then \( f \) and \( g \) are convolvable in the sense of the definition on page 171; moreover \( f * g \in {L}_{\mathrm{{loc}}}^{1}\left( {\mathbb{R}}^{d}\right) \) an... | Proof. For every \( \varphi \in \mathcal{D}\left( {\mathbb{R}}^{d}\right) \) , \[ \iint \left| {f\left( {x - y}\right) }\right| \left| {g\left( y\right) }\right| \left| {\varphi \left( x\right) }\right| {dxdy} = \iint \left| {f\left( x\right) }\right| \left| {g\left( y\right) }\right| \left| {\varphi \left( {x + y}\rig... | Yes |
Proposition 2.11 (Associativity) Let \( \left( {{T}_{1},{T}_{2},{T}_{3}}\right) \) be a family of distributions on \( {\mathbb{R}}^{d} \) satisfying (C). The distributions \( \left( {{T}_{1} * {T}_{2}}\right) * {T}_{3} \) and \( {T}_{1} * \left( {{T}_{2} * {T}_{3}}\right) \) are well-defined and coincide. | Proof. By property 1 on page 326, the distributions \( {T}_{1} * {T}_{2} \) and \( {T}_{2} * {T}_{3} \) are well defined and, by Proposition 2.7,\n\n\( \operatorname{Supp}\left( {{T}_{1} * {T}_{2}}\right) \subset \operatorname{Supp}{T}_{1} + \operatorname{Supp}{T}_{2},\;\operatorname{Supp}\left( {{T}_{2} * {T}_{3}}\rig... | Yes |
Proposition 2.12 If \( \\left( {{T}_{1},\\ldots ,{T}_{n}}\\right) \) satisfies condition (C), we have\n\n\[ \n{D}_{j}\\left( {{T}_{1} * \\cdots * {T}_{n}}\\right) = {T}_{1} * \\cdots * {T}_{k - 1} * {D}_{j}{T}_{k} * {T}_{k + 1} * \\cdots * {T}_{n} \n\]\n\nfor all \( j \\in \\{ 1,\\ldots, d\\} \) and \( k \\in \\{ 1,\\l... | Proof. Note first that \( \\operatorname{Supp}{D}_{j}{T}_{k} \\subset \\operatorname{Supp}{T}_{k} \), so the two sides in the equality above are well defined (see property 1 on page 326). By associativity and commutativity, it suffices to show that, if \( \\left( {T, S}\\right) \) satisfies \( \\left( \\mathrm{C}\\righ... | Yes |
Consider \( T \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{d}\right) \) and \( f \in \mathcal{E}\left( {\mathbb{R}}^{d}\right) \), and suppose \( \left( {T, f}\right) \) satisfies condition (C). Then \( T * f \in \mathcal{E}\left( {\mathbb{R}}^{d}\right) \) and, for all \( x \in {\mathbb{R}}^{d} \), the intersection... | Proof. For each \( l > 0 \), we again fix an element \( {\rho }_{l} \) of \( \mathcal{D}\left( {\mathbb{R}}^{d}\right) \) equal to 1 on \( B\left( {0, l}\right) \) . Take \( T \in {\mathcal{D}}^{\prime m}\left( {\mathbb{R}}^{d}\right) \) and \( f \in {\mathcal{E}}^{m + r}\left( {\mathbb{R}}^{d}\right) \) (or \( T \in {... | Yes |
Proposition 2.14 For every open \( \Omega \) in \( {\mathbb{R}}^{d} \), the set \( \mathcal{D}\left( \Omega \right) \) is dense in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) . In other words, every distribution on \( \Omega \) is the limit in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) of a sequence of e... | Proof. Let \( \Omega \) be open in \( {\mathbb{R}}^{d} \), and let \( {\left( {K}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence of compact sets exhausting \( \Omega \) . For every \( n \in \mathbb{N} \), take \( {\varphi }_{n} \in \mathcal{D}\left( \Omega \right) \) such that \( {\varphi }_{n} = 1 \) on \( {K}_{n} \... | Yes |
Proposition 3.1 If \( T \in {\mathcal{E}}^{\prime }\left( {\mathbb{R}}^{d}\right) \), then\n\n\[ T = \frac{1}{{s}_{d}}\mathop{\sum }\limits_{{j = 1}}^{d}\left( \frac{{x}_{j}}{{r}^{d}}\right) * {D}_{j}T \]\n\nwhere \( r = \left| x\right| \) and \( {s}_{d} \) is the area of the unit sphere in \( {\mathbb{R}}^{d} \) . | Proof. Let \( E \) be the fundamental solution of the Laplacian given in Theorem 3.2 on page 308. A simple calculation using Theorem 2.12 on page 301 shows that\n\n\[ {D}_{j}E = \frac{1}{{s}_{d}}\frac{{x}_{j}}{{r}^{d}}\;\text{ for all }j \in \{ 1,\ldots, d\} \]\n\nand this in any dimension \( d \) . At the same time, \... | Yes |
Proposition 3.2 The norm \( \parallel \cdot {\parallel }_{1, p} \) makes \( {W}^{1, p}\left( {\mathbb{R}}^{d}\right) \) into a Banach space. | Proof. Let \( {\left( {f}_{n}\right) }_{n \in \mathbb{N}} \) be a Cauchy sequence in \( {W}^{1, p}\left( {\mathbb{R}}^{d}\right) \) . Since the space \( {L}^{p}\left( {\mathbb{R}}^{d}\right) \) is complete, the sequences \( \left( {f}_{n}\right) ,\left( {{D}_{1}{f}_{n}}\right) ,\ldots ,\left( {{D}_{d}{f}_{n}}\right) \)... | Yes |
Theorem 3.4 Let \( T \) be a distribution on an open set \( \Omega \) in \( {\mathbb{R}}^{d} \) . Suppose that \( p \in \left\lbrack {1,\infty }\right\rbrack \) and that \( {D}_{j}T \in {L}_{\text{loc }}^{p}\left( \Omega \right) \) for every \( j \in \{ 1,\ldots, d\} \) . - If \( p \leq d \), then \( T \in {L}_{\text{l... | Proof. Let \( K \) be a compact subset of \( \Omega \) and \( {K}^{\prime } \) a compact subset of \( \Omega \) whose interior contains \( K \) . Let \( \varphi \in \mathcal{D}\left( \Omega \right) \) be such that \( \varphi = 1 \) on \( {K}^{\prime } \) . Put \( \mathfrak{d} = d\left( {K,\Omega \smallsetminus {\mathri... | Yes |
Theorem 3.5 Any differential operator with constant coefficients having a fundamental solution whose restriction to \( {\mathbb{R}}^{d} \smallsetminus \{ 0\} \) is a function of class \( {C}^{\infty } \) is hypoelliptic. | Proof. The proof is analogous to that of Theorem 3.4. Let \( \Omega \) be an open subset of \( {\mathbb{R}}^{d} \) and let \( K,{K}^{\prime } \) be compact subsets of \( \Omega \) such that \( K \subset {K}^{\prime } \) . Write \( \mathfrak{d} = d\left( {K,{\mathbb{R}}^{d} \smallsetminus {\mathring{K}}^{\prime }}\right... | Yes |
Consider a linear differential operator \( P\left( D\right) \) with constant coefficients, a fundamental solution \( E \) of \( P\left( D\right) \), and \( S \in {\mathcal{E}}^{\prime }\left( {\mathbb{R}}^{d}\right) \). The distribution \( {T}_{0} = E * S \) satisfies \( P\left( D\right) {T}_{0} = S \). Moreover, the s... | If \( S \in {\mathcal{E}}^{\prime }\left( {\mathbb{R}}^{d}\right) \), Proposition 2.12 yields \( P\left( D\right) \left( {E * S}\right) = P\left( D\right) E * S = \delta * S = S \). Set \( U = T - E * S \). Clearly, \( P\left( D\right) T = S \) if and only if \( P\left( D\right) U = 0 \). | Yes |
Proposition 1.2 Suppose that \( \Omega = \left( {a, b}\right) \), with \( - \infty \leq a < b \leq + \infty \) . Every element \( f \) of \( {H}^{1}\left( \Omega \right) \) has a continuous representative on \( \Omega \) (still denoted by \( f \) ) that has finite limits at \( a \) and \( b \) . Moreover, if \( a = - \... | Proof. By Theorem 2.8 on page 297, every element of \( {H}^{1}\left( \Omega \right) \) has a continuous representative \( f \) satisfying, for \( \alpha \in \Omega \) ,\n\n\[ f\left( t\right) = f\left( \alpha \right) + {\int }_{\alpha }^{t}{f}^{\prime }\left( u\right) {du}\;\text{ for all }t \in \Omega . \]\n\n(*) \n\n... | Yes |
Proposition 1.3 Suppose that \( \Omega = \left( {a, b}\right) \), with \( - \infty < a < b < + \infty \) . Then \( {C}^{1}\left( \bar{\Omega }\right) \) is a dense subspace of \( {H}^{1}\left( \Omega \right) \) . | Proof. Clearly \( {C}^{1}\left( \left\lbrack {a, b}\right\rbrack \right) \) is a subspace of \( {H}^{1}\left( \Omega \right) \) . Consider an element of \( {H}^{1}\left( \Omega \right) \), having a continuous representative \( f \) . By the preceding proposition (and Theorem 2.8 on page 297), \( f \) has a continuous e... | Yes |
Proposition 1.5 The spaces \( {H}^{1}\left( {\mathbb{R}}^{d}\right) \) and \( {H}_{0}^{1}\left( {\mathbb{R}}^{d}\right) \) coincide. | Proof. We must show that \( \mathcal{D}\left( {\mathbb{R}}^{d}\right) \) is dense in \( {H}^{1}\left( {\mathbb{R}}^{d}\right) \) . Take \( \xi \in \mathcal{D}\left( {\mathbb{R}}^{d}\right) \) such that \( \xi \left( 0\right) = 1 \) . For each \( n \in {\mathbb{N}}^{ * } \), put \( {\xi }_{n}\left( x\right) = \xi \left(... | Yes |
Proposition 1.6 (Poincaré inequality) If \( \Omega \) is a bounded open set in \( {\mathbb{R}}^{d} \) (more generally, if one of the projections of \( \Omega \) on the coordinate axes is bounded), there exists a constant \( C \geq 0 \) depending only on \( \Omega \) and such that \[ \parallel u{\parallel }_{{L}^{2}\lef... | Proof. By denseness, we just have to show the inequality for every \( u \in \) \( \mathcal{D}\left( \Omega \right) \), that is, for every \( u \in \mathcal{D}\left( {\mathbb{R}}^{d}\right) \) such that \( \operatorname{Supp}u \subset \Omega \) . Suppose for example that the projection on \( \Omega \) onto the first fac... | Yes |
Corollary 1.7 Suppose that \( \Omega \) is a bounded open set (more generally, that one of the coordinate projections of \( \Omega \) is bounded). The map \[ u \mapsto \parallel u{\parallel }_{{H}_{0}^{1}\left( \Omega \right) } = \parallel \left| {\nabla u}\right| {\parallel }_{{L}^{2}\left( \Omega \right) } \] is a Hi... | Proof. If \( C \) is the constant that appears in the Poincaré inequality, we have, for every \( u \in {H}_{0}^{1}\left( \Omega \right) \) , \[ {\begin{Vmatrix}\left| \nabla u\right| \end{Vmatrix}}_{{L}^{2}\left( \Omega \right) }^{2} \leq \parallel u{\parallel }_{{L}^{2}\left( \Omega \right) }^{2} + {\begin{Vmatrix}\le... | Yes |
Proposition 1.8 For \( f \in {H}_{0}^{1}\left( \Omega \right) \), set\n\n\[ \n\widetilde{f} = \left\{ \begin{array}{ll} f & \text{ on }\Omega , \\ 0 & \text{ on }{\mathbb{R}}^{d} \smallsetminus \Omega . \end{array}\right.\n\]\n\nThen \( \widetilde{f} \in {H}^{1}\left( {\mathbb{R}}^{d}\right) \) and the map that takes \... | Proof. If \( f \in \mathcal{D}\left( \Omega \right) \), we clearly have \( \widetilde{f} \in \mathcal{D}\left( {\mathbb{R}}^{d}\right) \) and \( \widetilde{{D}_{j}f} = {D}_{j}\widetilde{f} \) for \( j \in \{ 1,\ldots, d\} \) . Consequently, the map \( f \mapsto \widetilde{f} \) is an isometry from \( \mathcal{D}\left( ... | Yes |
Lemma 1.9 For every \( u \in {H}^{1}\left( {\mathbb{R}}^{d}\right) \) and every \( h \in {\mathbb{R}}^{d} \), \[ {\begin{Vmatrix}{\tau }_{h}u - u\end{Vmatrix}}_{{L}^{2}} \leq \parallel \left| {\nabla u}\right| {\parallel }_{{L}^{2}}\left| h\right| \] | Proof. By Proposition 1.5, the space \( \mathcal{D}\left( {\mathbb{R}}^{d}\right) \) is dense in \( {H}^{1}\left( {\mathbb{R}}^{d}\right) \). Thus it suffices to prove the property for \( u \in \mathcal{D}\left( {\mathbb{R}}^{d}\right) \). If \( u \in \mathcal{D}\left( {\mathbb{R}}^{d}\right) \), we have \[ u\left( {x ... | Yes |
Theorem 1.10 (Rellich) If \( \Omega \) is a bounded open set, the canonical injection \( u \mapsto u \) from \( {H}_{0}^{1}\left( \Omega \right) \) into \( {L}^{2}\left( \Omega \right) \) is a compact operator. | Proof. Since the map \( u \mapsto {u}_{\mid \Omega } \) from \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) to \( {L}^{2}\left( \Omega \right) \) is clearly continuous, it suffices to prove that the map \( f \mapsto \widetilde{f} \) from \( {H}_{0}^{1}\left( \Omega \right) \) to \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \)... | Yes |
Proposition 2.1 If \( f \in {L}^{2}\left( \Omega \right) \), these statements are equivalent:\n\n- \( u \in {H}_{0}^{1}\left( \Omega \right) \) and \( {\Delta u} = f \) .\n\n- \( u \in {H}_{0}^{1}\left( \Omega \right) \) and \( {\left( u \mid v\right) }_{{H}_{0}^{1}} = - {\left( f \mid v\right) }_{{L}^{2}} \) for all \... | Proof. If \( f \in {L}^{2}\left( \Omega \right) \) and \( u \in {H}_{0}^{1}\left( \Omega \right) \), we have the following chain of equivalences:\n\n\[ \n{\Delta u} = f \Leftrightarrow \langle {\Delta u},\bar{\varphi }\rangle = {\left( f \mid \varphi \right) }_{{L}^{2}}\;\text{ for all }\varphi \in \mathcal{D}\left( \O... | Yes |
For every \( f \in {L}^{2}\left( \Omega \right) \), the Dirichlet problem on \( \Omega \) with right-hand side \( f \) has a unique solution \( u \in {H}_{0}^{1}\left( \Omega \right) \). The operator \[ {\Delta }^{-1} : {L}^{2}\left( \Omega \right) \rightarrow {H}_{0}^{1}\left( \Omega \right) \] \[ f\; \mapsto \;u \] t... | Proof. If \( f \in {L}^{2}\left( \Omega \right) , \) \[ \left| {\left( f \mid v\right) }_{{L}^{2}}\right| \leq C\parallel f{\parallel }_{{L}^{2}}\parallel v{\parallel }_{{H}_{0}^{1}}\;\text{ for all }v \in {H}_{0}^{1}\left( \Omega \right) , \] where \( C \) is the constant in the Poincaré inequality. Thus, the map \( L... | Yes |
Proposition 2.3 Let \( f \in {L}^{2}\left( \Omega \right) \) . For every \( v \in {H}_{0}^{1}\left( \Omega \right) \), put\n\n\[ \n{J}_{f}\left( v\right) = \frac{1}{2}{\left( \parallel v{\parallel }_{{H}_{0}^{1}}\right) }^{2} + \operatorname{Re}{\left( f \mid v\right) }_{{L}^{2}}.\n\]\n\nThese statements are equivalent... | Proof. Suppose \( h \in {H}_{0}^{1}\left( \Omega \right) \) . Then\n\n\[ \n{J}_{f}\left( {u + h}\right) = {J}_{f}\left( u\right) + \mathrm{{Re}}\left( {{\left( f \mid h\right) }_{{L}^{2}} + {\left( u \mid h\right) }_{{H}_{0}^{1}}}\right) + \frac{1}{2}{\left( \parallel h{\parallel }_{{H}_{0}^{1}}\right) }^{2}.\n\]\n\nTh... | Yes |
Proposition 2.4 The operator\n\n\\[ \nT : {H}_{0}^{1}\\left( \\Omega \\right) \\rightarrow {H}_{0}^{1}\\left( \\Omega \\right) \n\\]\n\n\\[ \nv\\; \\mapsto \\;u\\text{ such that }{\\Delta u} = - v \n\\]\nis an injective, compact, positive selfadjoint operator on \\( {H}_{0}^{1}\\left( \\Omega \\right) \\) . | Proof. Let \\( J : u \\mapsto u \\) be the canonical injection from \\( {H}_{0}^{1}\\left( \\Omega \\right) \\) into \\( {L}^{2}\\left( \\Omega \\right) \\) . Then \\( T = - {\\Delta }^{-1} \\circ J \\), so, by Proposition 1.2 on page 215, \\( T \\) is compact, because \\( {\\Delta }^{-1} \\) is continuous (Theorem 2.2... | Yes |
Proposition 2.6 Suppose \( f \in {L}^{2}\left( \Omega \right) \) . The solution \( u \) of the Dirichlet problem on \( \Omega \) with right-hand side \( f \) is given by\n\n\[ u = - \mathop{\sum }\limits_{{n = 0}}^{{+\infty }}{\left( f \mid {u}_{n}\right) }_{{L}^{2}}{u}_{n} \]\n\nthe series being convergent in \( {H}_{... | Proof. By remark 1 above, \( {\left( \sqrt{-{\mu }_{n}}{u}_{n}\right) }_{n \in \mathbb{N}} \) is a Hilbert basis of \( {L}^{2}\left( \Omega \right) \), so\n\n\[ f = - \mathop{\sum }\limits_{{n = 0}}^{{+\infty }}{\mu }_{n}{\left( f \mid {u}_{n}\right) }_{{L}^{2}}{u}_{n} \]\n\nwith convergence in \( {L}^{2}\left( \Omega ... | Yes |
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