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Example 5.8.6 As a special case of 5.8.5 we see that every Borel set \( B \subseteq X \times Y \) with \( {B}_{x} \) a dense \( {G}_{\delta } \) set admits a Borel uniformization. However, there is an \( {F}_{\sigma } \) subset \( E \) of \( \left\lbrack {0,1}\right\rbrack \times {\mathbb{N}}^{\mathbb{N}} \) with secti... | Let \( C \subseteq \left\lbrack {0,1}\right\rbrack \times {\mathbb{N}}^{\mathbb{N}} \) be a closed set with projection to the first coordinate space \( \left\lbrack {0,1}\right\rbrack \), that does not admit a Borel uniformization. Such a set exists by 5.1.7. For each \( s \in {\mathbb{N}}^{ < \mathbb{N}} \), fix a hom... | No |
Theorem 5.8.7 (Blackwell and Ryll-Nardzewski [17]) Let \( X, Y \) be Polish spaces, \( P \) a transition probability on \( X \times Y \), and \( B \) a Borel subset of \( X \times Y \) such that \( P\left( {x,{B}_{x}}\right) > 0 \) for all \( x \in {\pi }_{X}\left( B\right) \) . Then \( {\pi }_{X}\left( B\right) \) is ... | Proof. Apply 5.8.4 with \( {\mathcal{I}}_{x} \) as in Example 5.8.1. | No |
Theorem 5.8.8 (Blackwell and Ryll-Nardzewski) Let \( X, Y \) be Polish spaces, \( \mathcal{A} \) a countably generated sub \( \sigma \) algebra of \( {\mathcal{B}}_{X} \), and \( P \) a transition probability on \( X \times Y \) such that for every \( B \in {\mathcal{B}}_{Y}, x \rightarrow P\left( {x, B}\right) \) is \... | Proof of 5.8.8. By a slight modification of the argument contained in the proof of 3.4.24 we see that for every \( E \in \mathcal{A} \otimes {\mathcal{B}}_{Y}, x \rightarrow P\left( {x,{E}_{x}}\right) \) is \( \mathcal{A} \) -measurable. As \( {\pi }_{X}\left( B\right) = \left\{ {x \in X : P\left( {x,{B}_{x}}\right) > ... | Yes |
Lemma 5.8.9 Let \( X, Y,\mathcal{A} \), and \( P \) be as above. For every \( E \in \mathcal{A}\bigotimes {\mathcal{B}}_{Y} \) and every \( \epsilon > 0 \), there is an \( F \in \mathcal{A}\bigotimes {\mathcal{B}}_{Y} \) contained in \( E \) such that \( {F}_{x} \) is compact and \( P\left( {x,{F}_{x}}\right) \geq \eps... | Proof. Let \( \mathcal{M} \) be the class of all sets in \( \mathcal{A}\bigotimes {\mathcal{B}}_{Y} \) such that the conclusion of the lemma holds for every \( P \) and every \( \epsilon > 0 \) . By 3.4.20, \( \mathcal{M} \) contains all rectangles \( A \times B \), where \( A \in \mathcal{A} \) and \( B \) Borel in \(... | Yes |
Proposition 5.8.10 Let \( X, f \), and \( \mathcal{A} \) be as above. An everywhere proper conditional distribution given \( f \) exists if and only if there is an \( \mathcal{A} \) - measurable \( g : X \rightarrow X \) such that \( f\left( {g\left( x\right) }\right) = f\left( x\right) \) for all \( x \) . | Proof. Suppose an \( \mathcal{A} \) -measurable \( g : X \rightarrow X \) such that \( f \circ g \) is the identity exists. Define\n\n\[ Q\left( {x, B}\right) = \left\{ \begin{array}{ll} 1 & \text{ if }g\left( x\right) \in B \\ 0 & \text{ otherwise. } \end{array}\right. \]\n\nIt is easy to verify that \( Q \) has the d... | Yes |
Proposition 5.8.13 (Feldman and Moore [41]) Every Borel equivalence relation on a Polish space \( X \) with equivalence classes countable is induced by a countable group of Borel automorphisms. | Proof. Let \( \Pi \) be a Borel equivalence relation on \( X \) with equivalence classes countable. By 5.8.11, write\n\n\[ \Pi = \mathop{\bigcup }\limits_{n}{G}_{n} \]\n\nwhere \( {\pi }_{1} \mid {G}_{n} \) is one-to-one, \( {\pi }_{1}\left( {x, y}\right) = x \) ; i.e., the \( {G}_{n} \) ’s are graphs of Borel function... | Yes |
Theorem 5.9.1 (Miller [85]) Every partition \( \Pi \) of a Polish space \( X \) into \( {G}_{\delta } \) sets such that the saturation of every basic open set is simultaneously \( {F}_{\sigma } \) and \( {G}_{\delta } \) admits a section \( s : X \rightarrow X \) that is Borel measurable of class 2. In particular, such... | Proof. Let \( \left( {U}_{n}\right) \) be a countable base for the topology of \( X \) . Let \( \left( {V}_{n}\right) \) be an enumeration of \( \left\{ {{U}_{n}^{ * } : n \in \mathbb{N}}\right\} \bigcup \left\{ {{\left( {U}_{n}^{ * }\right) }^{c} : n \in \mathbb{N}}\right\} \) . Let \( {\mathcal{T}}^{\prime } \) be th... | Yes |
Theorem 5.9.2 (Srivastava [114]) Every Borel measurable partition \( \mathbf{\Pi } \) of a Polish space \( X \) into \( {G}_{\delta } \) sets admits a Borel cross section. | Proof. (Kechris) For \( x \in X \) let \( \left\lbrack x\right\rbrack \) denote the member of \( \mathbf{\Pi } \) containing \( x \) . Consider the multifunction \( p : X \rightarrow X \) defined by\n\n\[ p\left( x\right) = \operatorname{cl}\left( \left\lbrack x\right\rbrack \right) \]\n\nThen \( p : X \rightarrow X \)... | Yes |
Theorem 5.9.5 \( \operatorname{irr}\left( A\right) / \sim \) is standard Borel if and only if \( A \) is GCR. | Its proof makes crucial uses of 5.4.3 and 4.5.4. We refer the interested reader to [4] and [43] for a proof. | No |
Theorem 5.10.1 (The reflection theorem) Let \( X \) be a Polish space and \( \Phi \subseteq \mathcal{P}\left( X\right) {\mathbf{\Pi }}_{1}^{1} \) on \( {\mathbf{\Pi }}_{1}^{1} \) . For every \( {\mathbf{\Pi }}_{1}^{1} \) set \( A \in \Phi \) there is a Borel \( B \subseteq A \) in \( \Phi \) . | Proof. Suppose there is a \( {\mathbf{\Pi }}_{1}^{1} \) set \( A \subseteq X \) in \( \Phi \) that does not contain a Borel set belonging to \( \Phi \) . We shall get a contradiction. Let \( \varphi \) be a \( {\mathbf{\Pi }}_{1}^{1} \) -norm on \( A \) and\n\n\[ C = \left\{ {\left( {x, y}\right) : y{ < }_{\varphi }^{ ... | Yes |
Theorem 5.10.2 Let \( X, Y \) be Polish spaces and \( A \subseteq X \times Y \) analytic with sections \( {A}_{x} \) countable. Then every coanalytic set \( B \) containing \( A \) contains a Borel set \( E \supseteq A \) with all sections countable. | Proof. Let \( C = {B}^{c} \) . Define \( \Phi \subseteq \mathcal{P}\left( {X \times Y}\right) \) by\n\n\[ D \in \Phi \Leftrightarrow {D}^{c} \subseteq B\& \forall x\left( {\left( {D}^{c}\right) }_{x}\right. \text{is countable})\text{.}\]\n\nUsing 4.3.7 we can easily check that \( \Phi \) is \( {\mathbf{\Pi }}_{1}^{1} \... | Yes |
Theorem 5.10.3 (Lusin) Every analytic set with countable sections, in the product of two Polish spaces, can be covered by countably many Borel graphs. | Proof. The result immediately follows from 5.10.2 and 5.8.11. | No |
Proposition 5.10.4 (Burgess) Let \( X \) be Polish, \( E \) an analytic equivalence relation on \( X \), and \( C \subseteq X \times X \) a coanalytic set containing \( E \) . Then there is a Borel equivalence relation \( B \) such that \( E \subseteq B \subseteq C \) . | Proof of 5.10.4. Applying 5.10.5 repeatedly, by induction on \( n \) we can define a sequence of Borel sets \( \left( {B}_{n}\right) \) such that\n\n\[ E \subseteq {B}_{n} \subseteq \mathcal{E}\left( {B}_{n}\right) \subseteq {B}_{n + 1} \subseteq C \]\n\nfor all \( n \) . Take \( B = \mathop{\bigcup }\limits_{n}{B}_{n}... | Yes |
Lemma 5.10.5 Let \( X \) be a Polish space, \( P \) analytic, \( C \) coanalytic, and \( \mathcal{E}\left( P\right) \subseteq C \) . Then there is a Borel set \( B \) containing \( P \) such that\n\n\[ \mathcal{E}\left( B\right) \subseteq C\text{.} \] | Proof. Define \( \Phi \subseteq \mathcal{P}\left( {X \times X}\right) \) by\n\n\[ D \in \Phi \Leftrightarrow \mathcal{E}\left( {D}^{c}\right) \subseteq C. \]\n\n\( \Phi \) is \( {\mathbf{\Pi }}_{1}^{1} \) on \( {\mathbf{\Pi }}_{1}^{1} \) . Further, \( {P}^{c} \in \Phi \) . By the reflection theorem (5.10.1), there is a... | Yes |
Corollary 5.10.6 For every analytic equivalence relation \( E \) on a Polish space \( X \) there exist Borel equivalence relations \( {B}_{\alpha },\alpha < {\omega }_{1} \), such that \( E = \) \( \mathop{\bigcap }\limits_{{\alpha < {\omega }_{1}}}{B}_{\alpha }. \) | Proof. By 4.3.17, write \( E = \mathop{\bigcap }\limits_{{\alpha < {\omega }_{1}}}{C}_{\alpha },{C}_{\alpha } \) coanalytic. By 5.10.4, for each \( \alpha \) there exists a Borel equivalence relation \( {B}_{\alpha } \) such that \( E \subseteq {B}_{\alpha } \subseteq {C}_{\alpha } \) . | Yes |
Theorem 5.11.4 Every countably generated sub \( \sigma \) -algebra of the Borel \( \sigma \) - algebra of a Polish space has a minimal complement. | Proof of 5.11.4. Let \( X \) be Polish and \( \mathcal{C} \) a countably generated sub \( \sigma \) -algebra of \( {\mathcal{B}}_{X} \) . \n\nCase 1. There is a cocountable atom \( A \) of \( \mathcal{C} \) .\n\nLet \( f : X \smallsetminus A \rightarrow A \) be a one-to-one map. Take\n\n\[ \mathcal{D} = \sigma \left( {... | Yes |
Lemma 5.11.5 Let \( X \) be Polish and \( \mathcal{C} \) a countably generated sub \( \sigma \) -algebra of \( {\mathcal{B}}_{X} \) . Suppose \( \mathcal{D} \) is a countably generated sub \( \sigma \) -algebra of \( {\mathcal{B}}_{X} \) such that every atom \( A \) of \( \mathcal{D} \) is a partial cross section of th... | Proof. Under the hypothesis, \( \mathcal{C} \vee \mathcal{D} \) is a countably generated sub \( \sigma \) - algebra of \( {\mathcal{B}}_{X} \) with atoms singletons. Hence, by 4.5.7, \( \mathcal{C} \vee \mathcal{D} = {\mathcal{B}}_{X} \) . Let \( {\mathcal{D}}^{ * } \) be a proper countably generated sub \( \sigma \) -... | Yes |
Theorem 5.12.1 (Arsenin, Kunugui [60]) Let \( B \subseteq X \times Y \) be a Borel set, \( X, Y \) Polish, such that \( {B}_{x} \) is \( \sigma \) -compact for every \( x \) . Then \( {\pi }_{X}\left( B\right) \) is Borel, and \( B \) admits a Borel uniformization. | Proof of 5.12.1. Write \( B = \mathop{\bigcup }\limits_{n}{B}_{n} \), the \( {B}_{n} \) ’s Borel with compact sections. That this can be done follows from 5.12.3. Then\n\n\[{\pi }_{X}\left( B\right) = \mathop{\bigcup }\limits_{n}{\pi }_{X}\left( {B}_{n}\right)\]\n\nSince the projection of a Borel set with compact secti... | Yes |
Theorem 5.12.2 (Saint Raymond [97]) Let \( X, Y \) be Polish spaces and \( A, B \subseteq X \times Y \) analytic sets. Assume that for every \( x \), there is a \( \sigma \) -compact set \( K \) such that \( {A}_{x} \subseteq K \subseteq {B}_{x}^{c} \) . Then there exists a sequence of Borel sets \( \left( {B}_{n}\righ... | Null | No |
Theorem 5.12.3 Let \( X, Y \) be Polish spaces and \( A \subseteq X \times Y \) a Borel set with sections \( {A}_{x} \) \(\sigma\) -compact. Then \( A = \mathop{\bigcup }\limits_{n}{B}_{n} \), where each \( {B}_{n} \) is Borel with \( {\left( {B}_{n}\right) }_{x} \) compact for all \( x \) and all \( n \) . | Proof. The result trivially follows from 5.12.2 by taking \( B = {A}^{c} \) . | Yes |
Proposition 5.12.4 Let \( B \subseteq X \times Y \) be a Borel set with sections \( {B}_{x} \) that are \( {G}_{\delta } \) sets in \( Y \) . Then there exist Borel sets \( {B}_{n} \) with open sections such that \( B = \mathop{\bigcap }\limits_{n}{B}_{n} \) . | Proof. Let \( Z \) be a compact metric space containing (a homeomorph of) \( Y \) . Then \( B \) is Borel in \( X \times Z \) with sections \( {G}_{\delta } \) sets (2.2.7). By 5.12.3, there exist Borel sets \( {C}_{n} \) in \( X \times Z \) with sections compact such that \( \left( {X \times Z}\right) \smallsetminus B... | Yes |
Corollary 5.12.5 Let \( B \subseteq X \times Y \) be a Borel set with sections \( {B}_{x} \) that are \( {F}_{\sigma } \) sets in \( Y \) . Then there exist Borel sets \( {B}_{n} \) with closed sections such that \( B = \mathop{\bigcup }\limits_{n}{B}_{n} \) | Null | No |
Proposition 5.12.6 Let \( X \) be a Polish space and \( \mathcal{B} \subseteq F\left( X\right) \) hereditary. Then \( {\Omega }_{{D}_{\mathcal{B}}} = {\mathcal{B}}_{\sigma } \cap F\left( X\right) \) . | Proof. Fix a closed set \( A \subseteq X \) and a countable base \( \left( {U}_{n}\right) \) for \( X \) .\n\nLet \( {D}^{\infty }\left( A\right) = \varnothing \) . Then\n\n\[ A = \mathop{\bigcup }\limits_{{\alpha < {\left| A\right| }_{D}}}\left( {{D}^{\alpha }\left( A\right) \smallsetminus {D}^{\alpha + 1}\left( A\rig... | Yes |
Proposition 5.12.7 Let \( X \) be Polish and \( D \) a derivative on \( X \) such that\n\n\[ \n\\{ \\left( {A, B}\\right) \\in F\\left( X\\right) \\times F\\left( X\\right) : A \\subseteq D\\left( B\\right) \\} \n\]\n\nis analytic. Then\n\n(i) \( {\\Omega }_{D} \) is coanalytic, and\n\n(ii) for every analytic \( \\math... | Proof. Assertion (i) follows from the following equivalence:\n\n\[ \nA \\notin {\\Omega }_{D} \\Leftrightarrow \\exists B\\left( {B \\neq \\varnothing \\& B \\subseteq A\\& B \\subseteq D\\left( B\\right) }\\right) .\n\]\n\n(The sets \( A \) and \( B \) are closed in \( X \).)\n\nSuppose (ii) is false for some analytic... | Yes |
Proposition 5.12.10 Let \( X \) and \( Y \) be Polish spaces and \( A, B \) two disjoint analytic subsets of \( X \times Y \) such that \( {A}_{x} \) is closed and nowhere dense for all \( x \) . Then there is a Borel \( C \subseteq X \times Y \) such that the sections \( {C}_{x} \) are closed and nowhere dense, and su... | Null | No |
Proposition 5.12.11 (i) (Hillard [48]) Let \( X \) and \( Y \) be Polish spaces and \( A, B \) disjoint analytic subsets of \( X \times Y \) . Assume that the sections \( {A}_{x} \) are meager in \( Y \) . Then there is a sequence \( \left( {C}_{n}\right) \) of Borel sets with sections nowhere dense such that\n\n\[ A \... | Null | No |
Theorem 5.12.12 (A. Louveau [66]) For every \( 1 \leq \alpha < {\omega }_{1},{\mathbf{\sum }}_{\alpha }^{ * } = {\mathcal{F}}_{\alpha } \) . | Null | No |
Theorem 5.13.1 (Lopez-Escobar) A subset \( A \) of \( {X}_{L} \) is invariant (with respect to the logic action) and Borel, if and only if there is a sentence \( \sigma \) of \( {L}_{{\omega }_{1}\omega } \) such that \( A = {A}_{\sigma } \) . | Proof. The sufficient part of this result is proved by induction on formulae of \( {L}_{{\omega }_{1}\omega } \) as follows:\n\nFor every formula \( \phi \left\lbrack {{v}_{0},{v}_{1},\ldots ,{v}_{k - 1}}\right\rbrack \), the set\n\n\[ \n{A}_{\phi, k} = \left\{ {\left( {x,{n}_{0},{n}_{1},\ldots ,{n}_{k - 1}}\right) : {... | No |
Theorem 5.13.2 (Becker - Kechris) Suppose a Polish group \( G \) acts continuously on a Polish space \( X \) and \( A \) is an invariant Borel subset of \( X \) . Then there is a finer Polish topology on \( X \) making \( A \) clopen such that the action still remains continuous. | Null | No |
Theorem 5.13.3 (Becker - Kechris) Suppose a Polish group \( G \) acts on a Polish space \( X \) and the action is Borel. Then there is a finer Polish topology on \( X \) making the action continuous. | Null | No |
Theorem 5.13.4 (Burgess) Suppose \( E \) is an analytic equivalence relation on a Polish space \( X \) . Then the number of equivalence classes is \( \leq {\aleph }_{1} \) or perfectly many. | Null | No |
Theorem 5.13.8 Topological Vaught conjecture holds if \( G \) is a locally compact Polish group. | Assuming 5.13.9, we prove 5.13.8 as follows: Let \( G \) be a locally compact Polish group acting continuously on a Polish space \( X \) . Write \( G = \mathop{\bigcup }\limits_{n}{K}_{n} \) , \( {K}_{n} \) compact. Then, for \( x, y \in X \) ,\n\n\[ \exists g \in G\left( {y = g \cdot x}\right) \Leftrightarrow \exists ... | Yes |
Theorem 5.13.9 Let \( E \) be an analytic equivalence relation on a Polish space \( X \) with all equivalence classes \( {F}_{\sigma } \) . Then the number of equivalence classes is \( \leq {\aleph }_{0} \) or perfectly many. | Proof of 5.13.9. Let \( X \) be a Polish space and \( E \) an analytic equivalence relation on \( X \) with all its equivalence classes \( {F}_{\sigma } \) sets. Further assume that there are uncountably many \( E \) -equivalence classes. Fix a countable base \( \left( {V}_{n}\right) \) for the topology of \( X \) . Le... | No |
Proposition 5.13.10 Suppose \( X \) is a Polish space and \( E \) an equivalence relation on \( X \) which is meager in \( {X}^{2} \) . Then \( E \) has perfectly many equivalence classes. | Proof. Let \( E \subseteq \mathop{\bigcup }\limits_{n}{F}_{n},{F}_{n} \) closed and nowhere dense in \( {X}^{2} \) . Without any loss of generality, we further assume that the diagonal \( \left\{ {\left( {x, y}\right) \in {X}^{2} : x = y}\right\} \) is contained in each of \( {F}_{n} \) .\n\nFor each \( s \in {2}^{ < \... | Yes |
Theorem 5.13.11 (Silver's theorem) Suppose \( E \) is a coanalytic equivalence relation on a Polish space \( X \) . Then the number of equivalence classes is countable or perfectly many. | Null | No |
Theorem 5.13.12 (Sami) Topological Vaught conjecture holds if \( G \) is abelian. | Proof. Assume that the number of orbits is uncountable. We shall show that there is a perfect set of inequivalent elements.\n\nLet \( E \) be the equivalence relation on \( X \) defined by\n\n\[ \n{xEy} \Leftrightarrow {G}_{x} = {G}_{y} \n\]\n\nwhere \( {G}_{x} \) is the stabilizer of \( x \) . Let \( y = g \cdot x \) ... | Yes |
Lemma 5.13.14 Suppose \( \left\{ {{A}_{\alpha } : \alpha < {\omega }_{1}}\right\} \) is a family of Borel subsets of a Polish space \( X \) and \( E \) the equivalence relation on \( X \) defined by\n\n\[ \n{xEy} \Leftrightarrow \forall \alpha \left( {x \in {A}_{\alpha } \Leftrightarrow y \in {A}_{\alpha }}\right), x, ... | Proof of 5.13.14. Although the proof of the lemma is messy looking, ideawise it is quite simple. Assume that the number of \( E \) -equivalence classes is \( > {\aleph }_{1} \) . We shall then show that there are perfectly many \( E \) -equivalence classes. The following fact will be used repeatedly in the proof of the... | Yes |
Example 5.13.15 Let \( L \) be a first order language whose non-logical symbols consists of exactly one binary relation symbol. So, \( {X}_{L} = {2}^{\omega \times \omega } \) . We claim that in this case the equivalence relation \( {E}_{a} \) induced by the logic action is not Borel. Suppose not. Then \( {E}_{a} \in {... | It follows that \( W{O}^{\alpha } = \{ x \in {WO} : \left| x\right| \leq \alpha \} \in {\mathbf{\sum }}_{\beta }^{0} \) for every \( \alpha < {\omega }_{1} \) . Now take any Borel set \( A \) in \( {\mathbb{N}}^{\mathbb{N}} \) which is not of additive class \( \beta \) . Since \( {WO} \) is \( {\mathbf{\Pi }}_{1}^{1} \... | Yes |
Theorem 5.13.18 (Stern) Let \( E \) be an analytic equivalence relation on a Polish space \( X \) such that all but countably many equivalence classes are \( {F}_{\sigma } \) or \( {G}_{\delta } \) . The the number of equivalence classes is \( \leq {\aleph }_{0} \) or perfectly many. | Null | No |
Theorem 5.13.19 (Stern) Assume analytic determinacy. Let \( E \) be an analytic equivalence relation on a Polish space \( X \) such that all but countably many equivalence classes are of bounded Borel rank. Then the number of equivalence classes is \( \leq {\aleph }_{0} \) or perfectly many. | The proof this result is beyond the scope of this book. | No |
Theorem 5.14.1 (Kondô’s theorem) Let \( X, Y \) be Polish spaces. Every coanalytic set \( C \subseteq X \times Y \) admits a coanalytic uniformization. | We shall show that there is a sequence of coanalytic norms on a given co-analytic set with certain \ | No |
Theorem 5.14.4 Every coanalytic subset of \( {\mathbb{N}}^{\mathbb{N}} \) admits a very good \( {\mathbf{\Pi }}_{1}^{1} \) - scale. | Null | No |
Corollary 5.14.5 Let \( X \) be a Polish space and \( A \subseteq X \) coanalytic. Then A admits a very good \( {\mathbf{\Pi }}_{1}^{1} \) -scale. | Proof. By 2.6.9 there is a closed set \( D \subseteq {\mathbb{N}}^{\mathbb{N}} \) and a continuous bijection \( f : D \rightarrow X \) . Now, \( {f}^{-1}\left( A\right) \cap D \) is a \( {\mathbf{\Pi }}_{1}^{1} \) subset of \( {\mathbb{N}}^{\mathbb{N}} \) and hence admits a very good \( {\mathbf{\Pi }}_{1}^{1} \) -scal... | Yes |
We consider the discretization of the boundary value problem for the ordinary differential equation\n\n\[ - {u}^{\prime \prime }\left( x\right) = f\left( {x, u\left( x\right) }\right) ,\;x \in \left\lbrack {0,1}\right\rbrack \]\n\n(2.1)\n\nwith boundary condition\n\n\[ u\left( 0\right) = u\left( 1\right) = 0. \]\n\n(2.... | For the approximate solution we choose an equidistant subdivision of the interval \( \left\lbrack {0,1}\right\rbrack \) by setting\n\n\[ {x}_{j} = {jh},\;j = 0,\ldots, n + 1, \]\n\nwhere the step size is given by \( h = 1/\left( {n + 1}\right) \) with \( n \in \mathbb{N} \). At the internal grid points \( {x}_{j}, j = ... | Yes |
Consider the linear integral equation\n\n\[ \varphi \left( x\right) - {\int }_{0}^{1}K\left( {x, y}\right) \varphi \left( y\right) {dy} = f\left( x\right) ,\;x \in \left\lbrack {0,1}\right\rbrack ,\] | For the numerical approximation we replace the integral by the rectangular sum\n\n\[ {\int }_{0}^{1}K\left( {x, y}\right) \varphi \left( y\right) {dy} \approx \frac{1}{n}\mathop{\sum }\limits_{{k = 1}}^{n}K\left( {x,{x}_{k}}\right) \varphi \left( {x}_{k}\right) \]\n\nwith equidistant grid points \( {x}_{k} = k/n, k = 1... | Yes |
Consider some (physical) quantity \( u \) depending on time \( t \) and a parameter vector \( a = {\left( {a}_{1},\ldots ,{a}_{n}\right) }^{T} \in {\mathbb{R}}^{n} \) in terms of a known function\n\n\[ u\left( t\right) = f\left( {t;a}\right) \]\n\nIn order to determine the values of the parameter \( a \) (representing ... | The necessary conditions for a minimum,\n\n\[ \frac{\partial g}{\partial {a}_{j}} = 0,\;j = 1,\ldots, n \]\n\nlead to the normal equations\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{m}\left\lbrack {u\left( {t}_{k}\right) - f\left( {{t}_{k};a}\right) }\right\rbrack \frac{\partial f\left( {{t}_{k};a}\right) }{\partial {a}_{j... | Yes |
We consider the system\n\n\[ \n{x}_{1} + {200}{x}_{2} = {100} \]\n\n\[ \n{x}_{1} + \;{x}_{2} = 1 \]\n\nwith the exact solution \( {x}_{1} = {100}/{199} = {0.502}\ldots ,{x}_{2} = {99}/{199} = {0.497}\ldots \) . | For the following computations we use two-decimal-digit floating-point arithmetic. Column pivoting leads to \( {a}_{11} \) as pivot element, and the elimination yields\n\n\[ \n{x}_{1} + {200}{x}_{2} = {100} \]\n\n\[ \n- {200}{x}_{2} = - {99} \]\n\nsince \( {199} = {200} \) in two-digit floating-point representation. Fr... | Yes |
Theorem 2.7 Gaussian elimination for the simultaneous solution of an \( n \times n \) system for \( r \) different right-hand sides requires a total of\n\n\[ \frac{{n}^{3}}{3} + r{n}^{2} - \frac{n}{3} \]\n\nmultiplications. | Null | No |
Theorem 2.9 For a nonsingular matrix \( A \), Gaussian elimination (without reordering rows and columns) yields an LR decomposition. | Proof. In the first elimination step we multiply the first equation by \( {a}_{j1}/{a}_{11} \) and subtract the result from the \( j \) th equation; i.e., the matrix \( {A}_{1} = A \) is multiplied from the left by the lower triangular matrix\n\n\[ \n{L}_{1} = \left( \begin{array}{rrrr} 1 & & & \\ - \frac{{a}_{21}}{{a}... | Yes |
Theorem 2.13 To each \( n \times n \) matrix a QR decomposition can be obtained through \( n - 1 \) Householder transformations. | Null | No |
Example 3.2 Some examples of norms on \( {\mathbb{R}}^{n} \) and \( {\mathbb{C}}^{n} \) are given by\n\n\[ \n\parallel x{\parallel }_{1} \mathrel{\text{:=}} \mathop{\sum }\limits_{{j = 1}}^{n}\left| {x}_{j}\right| ,\;\parallel x{\parallel }_{2} \mathrel{\text{:=}} {\left( \mathop{\sum }\limits_{{j = 1}}^{n}{\left| {x}_... | Null | No |
Theorem 3.5 The limit of a convergent sequence is uniquely determined. | Proof. Assume that \( {x}_{n} \rightarrow x \) and \( {x}_{n} \rightarrow y \) for \( n \rightarrow \infty \) . Then from the triangle inequality we obtain that\n\n\[ \parallel x - y\parallel = \begin{Vmatrix}{x - {x}_{n} + {x}_{n} - y}\end{Vmatrix} \leq \begin{Vmatrix}{x - {x}_{n}}\end{Vmatrix} + \begin{Vmatrix}{{x}_{... | Yes |
Theorem 3.7 Two norms \( \parallel \cdot {\parallel }_{a} \) and \( \parallel \cdot {\parallel }_{b} \) on a linear space \( X \) are equivalent if and only if there exist positive numbers \( c \) and \( C \) such that\n\n\[ c\parallel x{\parallel }_{a} \leq \parallel x{\parallel }_{b} \leq C\parallel x{\parallel }_{a}... | Proof. Provided that the conditions are satisfied, from \( {\begin{Vmatrix}{x}_{n} - x\end{Vmatrix}}_{a} \rightarrow 0 \) , \( n \rightarrow \infty \), it follows that \( {\begin{Vmatrix}{x}_{n} - x\end{Vmatrix}}_{b} \rightarrow 0, n \rightarrow \infty \), and vice versa.\n\nConversely, let the two norms be equivalent ... | Yes |
Theorem 3.8 On a finite-dimensional linear space all norms are equivalent. | Proof. In a linear space \( X \) with finite dimension \( n \) and basis \( {u}_{1},\ldots ,{u}_{n} \) every element can be expressed in the form\n\n\[ x = \mathop{\sum }\limits_{{j = 1}}^{n}{\alpha }_{j}{u}_{j} \]\n\nAs in Example 3.2,\n\n\[ \parallel x{\parallel }_{\infty } \mathrel{\text{:=}} \mathop{\max }\limits_{... | Yes |
Theorem 3.11 Any bounded sequence in a finite-dimensional normed space \( X \) contains a convergent subsequence. | Proof. Let \( {u}_{1},\ldots ,{u}_{n} \) be a basis of \( X \) and let \( \left( {x}_{\nu }\right) \) be a bounded sequence. Then writing\n\n\[ \n{x}_{\nu } = \mathop{\sum }\limits_{{j = 1}}^{n}{\alpha }_{j\nu }{u}_{j} \n\] \n\nand using the norm (3.2), as in the proof of Theorem 3.8 we deduce that each of the sequence... | Yes |
Theorem 3.14 For a scalar product we have the Cauchy-Schwarz inequality\n\n\[ \n{\left| \left( x, y\right) \right| }^{2} \leq \left( {x, x}\right) \left( {y, y}\right) \n\]\n\nfor all \( x, y \in X \), with equality if and only if \( x \) and \( y \) are linearly dependent. | Proof. The inequality is trivial for \( x = 0 \) . For \( x \neq 0 \) it follows from\n\n\[ \n\left( {{\alpha x} + {\beta y},{\alpha x} + {\beta y}}\right) = {\left| \alpha \right| }^{2}\left( {x, x}\right) + 2\operatorname{Re}\{ \alpha \bar{\beta }\left( {x, y}\right) \} + {\left| \beta \right| }^{2}\left( {y, y}\righ... | Yes |
Theorem 3.14 For a scalar product we have the Cauchy-Schwarz inequality\n\n\\[ \n{\\left| \\left( x, y\\right) \\right| }^{2} \\leq \\left( {x, x}\\right) \\left( {y, y}\\right) \n\\]\n\nfor all \\( x, y \\in X \\), with equality if and only if \\( x \\) and \\( y \\) are linearly dependent. | Proof. The inequality is trivial for \\( x = 0 \\) . For \\( x \\neq 0 \\) it follows from\n\n\\[ \n\\left( {{\\alpha x} + {\\beta y},{\\alpha x} + {\\beta y}}\\right) = {\\left| \\alpha \\right| }^{2}\\left( {x, x}\\right) + 2\\operatorname{Re}\\{ \\alpha \\bar{\\beta }\\left( {x, y}\\right) \\} + {\\left| \\beta \\ri... | Yes |
Theorem 3.15 A scalar product \( \left( {\cdot , \cdot }\right) \) on a linear space \( X \) defines a norm by\n\n\[ \parallel x\parallel \mathrel{\text{:=}} {\left( x, x\right) }^{1/2} \]\n\nfor all \( x \in X \) ; i.e., a pre-Hilbert space is always a normed space. | Proof. We leave it as an exercise for the reader to verify the norm axioms. The triangle inequality follows by\n\n\[ \parallel x + y{\parallel }^{2} = \left( {x + y, x + y}\right) \leq \parallel x{\parallel }^{2} + 2\parallel x\parallel \parallel y\parallel + \parallel y{\parallel }^{2} = {\left( \parallel x\parallel +... | No |
Theorem 3.17 The elements of an orthonormal system are linearly independent. | Proof. From\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{n}{\alpha }_{k}{q}_{k} = 0 \]\n\nfor the orthonormal system \( \left\{ {{q}_{1},\ldots ,{q}_{n}}\right\} \), by taking the scalar product with \( {q}_{j} \), we immediately have that \( {\alpha }_{j} = 0 \) for \( j = 1,\ldots, n \) . | Yes |
Theorem 3.18 Let \( \left\{ {{u}_{0},{u}_{1},\ldots }\right\} \) be a finite or countable number of linearly independent elements of a pre-Hilbert space. Then there exists a uniquely determined orthogonal system \( \left\{ {{q}_{0},{q}_{1},\ldots }\right\} \) of the form\n\n\[ \n{q}_{n} = {u}_{n} + {r}_{n},\;n = 0,1,\l... | Proof. Assume that we have constructed orthogonal elements of the form (3.3) with the property (3.4) up to \( {q}_{n - 1} \) . By (3.4), the \( \left\{ {{q}_{0},\ldots ,{q}_{n - 1}}\right\} \) are linearly independent, and therefore \( \begin{Vmatrix}{q}_{k}\end{Vmatrix} \neq 0 \) for \( k = 0,1,\ldots, n - 1 \) . Henc... | Yes |
Theorem 3.21 A linear operator is continuous if it is continuous at one element. | Proof. Let \( A : X \rightarrow Y \) be continuous at \( {x}_{0} \in X \) . Then for every \( x \in X \) and every sequence \( \left( {x}_{n}\right) \) with \( {x}_{n} \rightarrow x, n \rightarrow ∞ \), we have\n\n\[ A{x}_{n} = A\left( {{x}_{n} - x + {x}_{0}}\right) + A\left( {x - {x}_{0}}\right) \rightarrow A\left( {x... | Yes |
Theorem 3.23 A linear operator \( A : X \rightarrow Y \) is bounded if and only if\n\n\[ \parallel A\parallel \mathrel{\text{:=}} \mathop{\sup }\limits_{{\parallel x\parallel = 1}}\parallel {Ax}\parallel < \infty . \]\n\nThe number \( \parallel A\parallel \) is the smallest bound for \( A \) and is called the norm of \... | Proof. Assume that \( A \) is bounded with the bound \( C \) . Then\n\n\[ \mathop{\sup }\limits_{{\parallel x\parallel = 1}}\parallel {Ax}\parallel \leq C \]\n\nand, in particular, \( \parallel A\parallel \) is less than or equal to any bound for \( A \) . Conversely, if \( \parallel A\parallel < \infty \), then using ... | Yes |
Theorem 3.24 A linear operator is continuous if and only if it is bounded. | Proof. Let \( A : X \rightarrow Y \) be bounded and let \( \left( {x}_{n}\right) \) be a sequence in \( X \) with \( {x}_{n} \rightarrow 0, n \rightarrow \infty \) . Then from \( \begin{Vmatrix}{A{x}_{n}}\end{Vmatrix} \leq C\begin{Vmatrix}{x}_{n}\end{Vmatrix} \) it follows that \( A{x}_{n} \rightarrow 0 \) , \( n \righ... | Yes |
Theorem 3.27 To each matrix \( A \) there exists a unitary matrix \( Q \) such that \( {Q}^{ * }{AQ} \) is an upper triangular matrix. | Proof. Assume that it has been shown that for each \( \left( {n - 1}\right) \times \left( {n - 1}\right) \) matrix \( {A}_{n - 1} \) there exists a unitary \( \left( {n - 1}\right) \times \left( {n - 1}\right) \) matrix \( {Q}_{n - 1} \) such that \( {Q}_{n - 1}^{ * }{A}_{n - 1}{Q}_{n - 1} \) is an upper triangular mat... | Yes |
Lemma 3.28 For an \( n \times n \) matrix \( A \) and its adjoint \( {A}^{ * } \) we have that\n\n\[ \left( {{Ax}, y}\right) = \left( {x,{A}^{ * }y}\right) \]\n\nfor all \( x, y \in {\mathbb{C}}^{n} \), where \( \left( {\cdot , \cdot }\right) \) denotes the Euclidean scalar product. | Proof. Simple calculations yield\n\n\[ \left( {{Ax}, y}\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{\left( Ax\right) }_{j}{\bar{y}}_{j} = \mathop{\sum }\limits_{{j = 1}}^{n}\mathop{\sum }\limits_{{k = 1}}^{n}{a}_{jk}{x}_{k}{\bar{y}}_{j} \]\n\n\[ = \mathop{\sum }\limits_{{k = 1}}^{n}\mathop{\sum }\limits_{{j = 1}}^{n}{... | Yes |
The eigenvalues of a Hermitian \( n \times n \) matrix are real, and the eigenvectors form an orthogonal basis in \( {\mathbb{C}}^{n} \) . | Proof. If \( A \) is Hermitian, i.e., if \( A = {A}^{ * } \), then the matrix \( \widetilde{A} \mathrel{\text{:=}} {Q}^{ * }{AQ} \) from Theorem 3.27 is also Hermitian, since\n\n\[ \n{\widetilde{A}}^{ * } = {\left( {Q}^{ * }AQ\right) }^{ * } = {Q}^{ * }{A}^{ * }{Q}^{* * } = {Q}^{ * }{AQ} = \widetilde{A}.\n\]\n\nTherefo... | Yes |
For an \( n \times n \) matrix \( A \) we have\n\n\[ \parallel A{\parallel }_{2} = \sqrt{\rho \left( {{A}^{ * }A}\right) } \]\n\nIf \( A \) is Hermitian, then\n\n\[ \parallel A{\parallel }_{2} = \rho \left( A\right) \] | Proof. From Lemma 3.28 we have that\n\n\[ \parallel {Ax}{\parallel }_{2}^{2} = \left( {{Ax},{Ax}}\right) = \left( {x,{A}^{ * }{Ax}}\right) \]\n\nfor all \( x \in {\mathbb{C}}^{n} \) . Hence the Hermitian matrix \( {A}^{ * }A \) is positive semidefinite and therefore has \( n \) orthonormal eigenvectors\n\n\[ {A}^{ * }A... | Yes |
Theorem 3.32 For each norm on \( {\mathbb{C}}^{n} \) and each \( n \times n \) matrix \( A \) we have that\n\n\[ \rho \left( A\right) \leq \parallel A\parallel \]\n\nConversely, to each matrix \( A \) and each \( \varepsilon > 0 \) there exists a norm on \( {\mathbb{C}}^{n} \) such that\n\n\[ \parallel A\parallel \leq ... | Proof. Let \( \lambda \) be an eigenvalue of \( A \) with eigenvector \( u \) . We may assume that \( \parallel u\parallel = 1 \) . Then the first part of the theorem follows from\n\n\[ \parallel A\parallel = \mathop{\sup }\limits_{{\parallel x\parallel = 1}}\parallel {Ax}\parallel \geq \parallel {Au}\parallel = \paral... | Yes |
Theorem 3.34 Every convergent sequence is a Cauchy sequence. | Proof. Let \( {x}_{n} \rightarrow x, n \rightarrow \infty \) . Then, for \( \varepsilon > 0 \) there exists \( N\left( \varepsilon \right) \in \mathbb{N} \) such that \( \begin{Vmatrix}{{x}_{n} - x}\end{Vmatrix} < \varepsilon /2 \) for all \( n \geq N\left( \varepsilon \right) \) . Now the triangle inequality yields\n\... | Yes |
Example 3.36 The linear space \( C\left\lbrack {a, b}\right\rbrack \) furnished with the maximum norm\n\n\[ \parallel f{\parallel }_{\infty } \mathrel{\text{:=}} \mathop{\max }\limits_{{x \in \left\lbrack {a, b}\right\rbrack }}\left| {f\left( x\right) }\right| \]\n\nis a Banach space. | Proof. The norm axioms (N1)-(N3) are trivially satisfied. The triangle inequality follows from\n\n\[ \parallel f + g{\parallel }_{\infty } = \mathop{\max }\limits_{{x \in \left\lbrack {a, b}\right\rbrack }}\left| {\left( {f + g}\right) \left( x\right) }\right| = \left| {\left( {f + g}\right) \left( {x}_{0}\right) }\rig... | Yes |
The linear space \( C\left\lbrack {a, b}\right\rbrack \) equipped with the \( {L}_{1} \) norm\n\n\[ \parallel f{\parallel }_{1} \mathrel{\text{:=}} {\int }_{a}^{b}\left| {f\left( x\right) }\right| {dx} \]\n\nis not complete. | The norm axioms are trivially satisfied. Without loss of generality we take \( \left\lbrack {a, b}\right\rbrack = \left\lbrack {0,2}\right\rbrack \) and choose\n\n\[ {f}_{n}\left( x\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} {x}^{n}, & 0 \leq x \leq 1 \\ 1, & 1 \leq x \leq 2 \end{array}\right. \]\n\nThen for ... | Yes |
Example 3.38 The linear space \( C\left\lbrack {a, b}\right\rbrack \) equipped with the \( {L}_{2} \) norm\n\n\[ \n\parallel f{\parallel }_{2} \mathrel{\text{:=}} {\left( {\int }_{a}^{b}{\left| f\left( x\right) \right| }^{2}dx\right) }^{1/2} \n\]\n\nis not complete. | Proof. The norm is generated by the scalar product\n\n\[ \n\left( {f, g}\right) \mathrel{\text{:=}} {\int }_{a}^{b}f\left( x\right) g\left( x\right) {dx}. \n\]\n\nConsidering the same sequence as in Example 3.37, it can be seen that \( C\left\lbrack {a, b}\right\rbrack \) also is not complete with respect to the \( {L}... | Yes |
Theorem 3.39 Each finite-dimensional normed space is a Banach space. | Proof. Let \( X \) be finite-dimensional with basis \( {u}_{1},\ldots ,{u}_{n} \) and assume that \( \left( {x}_{\nu }\right) \) is a Cauchy sequence in \( X \) . We represent\n\n\[ \n{x}_{\nu } = \mathop{\sum }\limits_{{j = 1}}^{n}{\alpha }_{j\nu }{u}_{j} \n\] \n\nand recall from Theorem 3.8 that there exists \( C > 0... | Yes |
Theorem 3.44 Each contraction operator has at most one fixed point. | Proof. Assume that \( x \) and \( y \) are two different fixed points of the contraction operator \( A \) . Then\n\n\[ 0 \neq \parallel x - y\parallel = \parallel {Ax} - {Ay}\parallel \leq q\parallel x - y\parallel \]\n\nwhence \( 1 \leq q \) follows. This is a contradiction to the fact that \( A \) is a contraction op... | Yes |
Theorem 3.45 (Banach) Let \( U \) be a complete subset of a normed space \( X \) and let \( A : U \rightarrow U \) be a contraction operator. Then \( A \) has a unique fixed point. | Proof. Starting from an arbitrary element \( {x}_{0} \in U \) we define a sequence \( \left( {x}_{n}\right) \) in \( U \) by the recursion\n\n\[ \n{x}_{n + 1} \mathrel{\text{:=}} A{x}_{n},\;n = 0,1,2,\ldots \n\]\n\nThen we have\n\n\[ \n\begin{Vmatrix}{{x}_{n + 1} - {x}_{n}}\end{Vmatrix} = \begin{Vmatrix}{A{x}_{n} - A{x... | Yes |
Theorem 3.46 Let \( A \) be a contraction operator with contraction constant \( q \) mapping a complete subset \( U \) of a normed space \( X \) into itself. Then the successive approximations\n\n\[ \n{x}_{n + 1} \mathrel{\text{:=}} A{x}_{n},\;n = 0,1,2,\ldots ,\n\]\n\nwith arbitrary \( {x}_{0} \in U \) converge to the... | Proof. The a priori error estimate follows from (3.12) by passing to the limit \( m \rightarrow \infty \) . The a posteriori estimate follows from the a priori estimate applied with starting element \( {x}_{0} = {x}_{n - 1} \) . | No |
Example 3.47 The function \( f : \lbrack 0,\infty ) \rightarrow \lbrack 0,\infty ) \) given by\n\n\[ f\left( x\right) \mathrel{\text{:=}} x + \frac{1}{1 + x} \]\n\nas a consequence of\n\n\[ f\left( x\right) - f\left( y\right) = \frac{x + y + {xy}}{1 + x + y + {xy}}\left( {x - y}\right) \]\n\nfulfills the condition\n\n\... | Null | No |
Theorem 3.48 Let \( B : X \rightarrow X \) be a bounded linear operator on a Banach space \( X \) with \( \parallel B\parallel < 1 \), and let \( I : X \rightarrow X \) denote the identity operator. Then \( I - B \) is bijective; i.e., for each \( z \in X \) the equation\n\n\[ x - {Bx} = z \]\n\nhas a unique solution \... | Proof. For fixed, but arbitrary, \( z \in X \) we define the operator \( A : X \rightarrow X \) by\n\n\[ {Ax} \mathrel{\text{:=}} {Bx} + z,\;x \in X. \]\n\nThen we have\n\n\[ \parallel {Ax} - {Ay}\parallel = \parallel B\left( {x - y}\right) \parallel \leq \parallel B\parallel \parallel x - y\parallel \]\n\nfor all \( x... | Yes |
Theorem 3.50 Let \( U \) be a finite-dimensional subspace of a normed space \( X \) . Then for every element in \( X \) there exists a best approximation with respect to \( U \) . | Proof. Let \( w \in X \) and choose a minimizing sequence \( \left( {u}_{n}\right) \) for \( w \) ; i.e., \( {u}_{n} \in U \) satisfies\n\n\[\n\begin{Vmatrix}{w - {u}_{n}}\end{Vmatrix} \rightarrow d \mathrel{\text{:=}} \mathop{\inf }\limits_{{u \in U}}\parallel w - u\parallel ,\;n \rightarrow \infty .\n\]\n\nBecause of... | Yes |
Theorem 3.51 Let \( U \) be a linear subspace of a pre-Hilbert space \( X \) . An element \( v \) is a best approximation to \( w \in X \) with respect to \( U \) if and only if\n\n\[ \left( {w - v, u}\right) = 0 \]\n\n(3.13)\n\nfor all \( u \in U \), i.e., if and only if \( w - v \bot U \) . To each \( w \in X \) ther... | Proof. We begin by noting the equality\n\n\[ \parallel w - u{\parallel }^{2} = \parallel w - v{\parallel }^{2} + 2\operatorname{Re}\left( {w - v, v - u}\right) + \parallel v - u{\parallel }^{2}, \]\n\n(3.14)\n\nwhich is valid for all \( u, v \in U \) . From this, sufficiency of the condition (3.13) is obvious, since \(... | Yes |
Theorem 3.52 Let \( U \) be a complete linear subspace of a pre-Hilbert space \( X \) . Then to each element \( w \in X \) there exists a unique best approximation with respect to \( U \) . The operator \( P : X \rightarrow U \) mapping \( w \in X \) onto its best approximation is a bounded linear operator with the pro... | Proof. Choose a sequence \( \left( {u}_{n}\right) \) with\n\n\[ \n{\begin{Vmatrix}w - {u}_{n}\end{Vmatrix}}^{2} \leq {d}^{2} + \frac{1}{n},\;n \in \mathbb{N},\n\]\n\n(3.15)\n\nwhere \( d \mathrel{\text{:=}} \mathop{\inf }\limits_{{u \in U}}\parallel w - u\parallel \) . Then\n\n\[ \n\parallel \left( {w - {u}_{n}}\right)... | Yes |
Corollary 3.53 Let \( U \) be a finite-dimensional linear subspace of a pre-Hilbert space \( X \) with basis \( {u}_{1},\ldots ,{u}_{n} \) . The linear combination\n\n\[ v = \mathop{\sum }\limits_{{k = 1}}^{n}{\alpha }_{k}{u}_{k} \]\n\nis the best approximation to \( w \in X \) with respect to \( U \) if and only if th... | Proof. The normal equations (3.16) obviously are equivalent to (3.13). | No |
Corollary 3.54 Let \( U \) be a finite-dimensional linear subspace of a pre-Hilbert space \( X \) with orthonormal basis \( {u}_{1},\ldots ,{u}_{n} \) . Then the orthogonal projection operator is given by\n\n\[ \n{Pw} = \mathop{\sum }\limits_{{k = 1}}^{n}\left( {w,{u}_{k}}\right) {u}_{k},\;w \in X.\n\] | Proof. This is trivial from either the orthogonality condition of Theorem 3.51 or the normal equations of Corollary 3.53. | Yes |
Theorem 4.1 Let \( B \) be an \( n \times n \) matrix. Then the successive approximations\n\n\[ \n{x}_{\nu + 1} \mathrel{\text{:=}} B{x}_{\nu } + z,\;\nu = 0,1,2,\ldots ,\n\]\n\nconverge for each \( z \in {\mathbb{C}}^{n} \) and each \( {x}_{0} \in {\mathbb{C}}^{n} \) if and only if\n\n\[ \n\rho \left( B\right) < 1\n\]... | Proof. If \( \rho \left( B\right) < 1 \), then by Theorem 3.32 there exists a norm \( \parallel \cdot \parallel \) on \( {\mathbb{C}}^{n} \) such that \( \parallel B\parallel < 1 \) . Now convergence follows from Theorem 3.48 together with the equivalence of all norms on \( {\mathbb{C}}^{n} \) according to Theorem 3.8.... | Yes |
Theorem 4.2 Assume that the matrix \( A = \left( {a}_{jk}\right) \) satisfies\n\n\[ \n{q}_{\infty } \mathrel{\text{:=}} \mathop{\max }\limits_{{j = 1,\ldots, n}}\mathop{\sum }\limits_{\substack{{k = 1} \\ {k \neq j} }}^{n}\left| \frac{{a}_{jk}}{{a}_{jj}}\right| < 1 \n\]\n\n(4.1)\n\nor\n\n\[ \n{q}_{1} \mathrel{\text{:=}... | Proof. The Jacobi matrix \( - {D}^{-1}\left( {{A}_{L} + {A}_{R}}\right) \) has diagonal entries zero and off-diagonal entries \( - {a}_{jk}/{a}_{jj} \) . Hence by Theorem 3.26 we have\n\n\[ \n{\begin{Vmatrix}-{D}^{-1}\left( {A}_{L} + {A}_{R}\right) \end{Vmatrix}}_{\infty } = {q}_{\infty } \n\]\n\n\[ \n{\begin{Vmatrix}-... | Yes |
Theorem 4.3 Assume that the matrix \( A = \left( {a}_{jk}\right) \) fulfills the Sassenfeld criterion\n\n\[ p \mathrel{\text{:=}} \mathop{\max }\limits_{{j = 1,\ldots, n}}{p}_{j} < 1 \]\n\nwhere the numbers \( {p}_{j} \) are recursively defined by\n\n\[ {p}_{1} \mathrel{\text{:=}} \mathop{\sum }\limits_{{k = 2}}^{n}\le... | Proof. Consider the equation\n\n\[ \left( {D + {A}_{L}}\right) x = - {A}_{R}z \]\n\nfor \( z \in {\mathbb{C}}^{n} \) with \( \parallel z{\parallel }_{\infty } = 1 \), that is,\n\n\[ {x}_{j} = - \mathop{\sum }\limits_{{k = 1}}^{{j - 1}}\frac{{a}_{jk}}{{a}_{jj}}{x}_{k} - \mathop{\sum }\limits_{{k = j + 1}}^{n}\frac{{a}_{... | Yes |
Corollary 4.4 Assume that the matrix \( A \) is strictly row-diagonally dominant. Then the Gauss-Seidel iterations converge. | Null | No |
Example 4.5 The tridiagonal matrix\n\n\[ A = \left( \begin{array}{rrrrrr} 2 & - 1 & & & & \\ - 1 & 2 & - 1 & & & \\ & - 1 & 2 & - 1 & & \\ & \cdot & \cdot & \cdot & \cdot & \cdot \\ & & & - 1 & 2 & - 1 \\ & & & & - 1 & 2 \end{array}\right) \] from Example 2.1 is not strictly row-diagonally dominant, but it satisfies th... | Proof. Obviously, \( {q}_{\infty } = 1 \) ; i.e.,(4.1) is not fulfilled. We have the recursion\n\n\[ {p}_{1} = \frac{1}{2},\;{p}_{j} = \frac{1}{2}{p}_{j - 1} + \frac{1}{2},\;j = 2,\ldots, n - 1,\;{p}_{n} = \frac{1}{2}{p}_{n - 1}. \]\n\nFrom this, by induction, it follows that\n\n\[ {p}_{j} = 1 - \frac{1}{{2}^{j}},\;j =... | Yes |
Assume that the Jacobi matrix \( B \mathrel{\text{:=}} - {D}^{-1}\left( {{A}_{L} + {A}_{R}}\right) \) has real eigenvalues and spectral radius less than one. Then the spectral radius of the iteration matrix\n\n\[ I - \omega {D}^{-1}A = \left( {1 - \omega }\right) I - \omega {D}^{-1}\left( {{A}_{L} + {A}_{R}}\right) \]\... | Proof. For \( \omega > 0 \) the equation \( {Bu} = {\lambda u} \) is equivalent to\n\n\[ \left\lbrack {\left( {1 - \omega }\right) I + {\omega B}}\right\rbrack u = \left\lbrack {1 - \omega + {\omega \lambda }}\right\rbrack u. \]\n\nHence the eigenvalues \( \lambda \) of \( B \) correspond to the eigenvalues \( 1 - \ome... | Yes |
Theorem 4.11 (Kahan) A necessary condition for the SOR method to be convergent is that \( 0 < \omega < 2 \) . | Proof. Since the eigenvalues \( {\mu }_{1},\ldots ,{\mu }_{n} \) of \( B\left( \omega \right) \) are the zeros of the characteristic polynomial, they satisfy\n\n\[ \mathop{\prod }\limits_{{j = 1}}^{n}{\mu }_{j} = \det B\left( \omega \right) \]\n\n(where multiple eigenvalues are repeated according to their algebraic mul... | Yes |
Theorem 4.12 (Ostrowski) If \( A \) is Hermitian and positive definite, then the SOR method converges for all \( {x}_{0} \in {\mathbb{C}}^{n} \), all \( y \in {\mathbb{C}}^{n} \), and all \( 0 < \omega < 2 \) to the unique solution of \( {Ax} = y \) . | Proof. Let \( \mu \) be an eigenvalue of \( B\left( \omega \right) \) with eigenvector \( x \) ; i.e., \[ \left\lbrack {\left( {1 - \omega }\right) D - \omega {A}_{R}}\right\rbrack x = \mu \left( {D + \omega {A}_{L}}\right) x. \] With the aid of \[ \left( {2 - \omega }\right) D - {\omega A} - \omega \left( {{A}_{R} - {... | Yes |
Corollary 4.16 Under the assumptions of Theorem 4.15 the Gauss-Seidel method converges twice as fast as the Jacobi method. | Proof. From (4.8) we observe that \( \mu = {\lambda }^{2} \) for \( \omega = 1 \) ; i.e., we have\n\n\[ \rho \left\lbrack {B\left( 1\right) }\right\rbrack = {\left\{ \rho \left\lbrack -{D}^{-1}\left( {A}_{L} + {A}_{R}\right) \right\rbrack \right\} }^{2} \]\n\nfor the spectral radii of the Gauss-Seidel matrix \( B\left(... | Yes |
For the tridiagonal matrix \( A \) from Example 4.5 we have\n\n\[ \frac{N\left( \mathrm{{SOR}}\right) }{N\left( \mathrm{{Jacobi}}\right) } \approx \frac{\pi }{4\left( {n + 1}\right) } \] \n\nfor the optimal relaxation parameter. | Proof. Using the trigonometric addition theorem\n\n\[ \frac{1}{2}\sin \frac{{\pi j}\left( {k - 1}\right) }{n + 1} + \frac{1}{2}\sin \frac{{\pi j}\left( {k + 1}\right) }{n + 1} = \cos \frac{\pi j}{n + 1}\sin \frac{\pi jk}{n + 1}, \] \n\nit can be seen that the Jacobi matrix\n\n\[ - {D}^{-1}\left( {{A}_{L} + {A}_{R}}\rig... | Yes |
Theorem 4.18 For the spectral radius of \( T \) we have that \( \rho \left( T\right) = {0.5} \) ; i.e., the two-grid iterations converge. | Proof. We note that from (4.18) and (4.19), with \( h \) replaced by \( {2h} \), we have\n\nthat\n\[ \n{A}^{\left( 2h\right) }{v}_{j}^{\left( 2h\right) } = \frac{1}{{h}^{2}}{\sin }^{2}\left( {\pi jh}\right) {v}_{j}^{\left( 2h\right) } = \frac{4}{{h}^{2}}{c}_{j}^{2}{s}_{j}^{2}{v}_{j}^{\left( 2h\right) }, \n\]\n\nwhence\... | Yes |
We consider the best approximation of a given continuous function \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \) by a polynomial \[ p\left( x\right) = \mathop{\sum }\limits_{{k = 0}}^{n}{\alpha }_{k}{x}^{k} \] of degree \( n \) in the least squares sense, i.e., with respect to the \( {L}_{2} \) norm. | Using the monomials \( x \mapsto {x}^{k}, k = 0,1,\ldots, n \), as a basis of the subspace \( {P}_{n} \subset C\left\lbrack {0,1}\right\rbrack \) of polynomials of degree less than or equal to \( n \) (see Theorem 8.2), from Corollary 3.53 and the integrals \[ {\int }_{0}^{1}{x}^{j}{x}^{k}{dx} = \frac{1}{j + k + 1} \] ... | Yes |
Theorem 5.3 Let \( X \) and \( Y \) be Banach spaces, let \( A : X \rightarrow Y \) be a bounded linear operator with a bounded inverse \( {A}^{-1} : Y \rightarrow X \) and let \( {A}^{\delta } : X \rightarrow Y \) be a bounded linear operator such that \( \begin{Vmatrix}{A}^{-1}\end{Vmatrix}\begin{Vmatrix}{{A}^{\delta... | Proof. Writing \( {A}^{\delta } = A\left\lbrack {I + {A}^{-1}\left( {{A}^{\delta } - A}\right) }\right\rbrack \), by Theorem 3.48 we observe that the inverse operator \( {\left\lbrack {A}^{\delta }\right\rbrack }^{-1} = {\left\lbrack I + {A}^{-1}\left( {A}^{\delta } - A\right) \right\rbrack }^{-1}{A}^{-1} \) exists and... | Yes |
Theorem 5.4 Let \( A \) be an \( m \times n \) matrix of rank \( r \) . Then there exist nonnegative numbers\n\n\[ \n{\mu }_{1} \geq {\mu }_{2} \geq \cdots \geq {\mu }_{r} > {\mu }_{r + 1} = \cdots = {\mu }_{n} = 0 \n\]\n\nand orthonormal vectors \( {u}_{1},\ldots ,{u}_{n} \in {\mathbb{C}}^{n} \) and \( {v}_{1},\ldots ... | Proof. The Hermitian and semipositive definite matrix \( {A}^{ * }A \) of rank \( r \) has \( n \) orthonormal eigenvectors \( {u}_{1},\ldots ,{u}_{n} \) with nonnegative eigenvalues\n\n\[ \n{A}^{ * }A{u}_{j} = {\mu }_{j}^{2}{u}_{j},\;j = 1,\ldots, n \n\]\n\n(5.7)\n\nwhich we may assume to be ordered according to \( {\... | Yes |
Theorem 5.5 Let \( A \) be an \( m \times n \) matrix of rank \( r \) with singular system \( \left( {{\mu }_{j},{u}_{j},{v}_{j}}\right) \). The linear system\n\n\[ \n{Ax} = y \n\]\n\n(5.9)\n\nis solvable if and only if\n\n\[ \n\left( {y, z}\right) = 0 \n\]\n\n\( \left( {5.10}\right) \)\n\nfor all \( z \in {\mathbb{C}}... | Proof. Let \( x \) be a solution of (5.9) and let \( {A}^{ * }z = 0 \). Then\n\n\[ \n\left( {y, z}\right) = \left( {{Ax}, z}\right) = \left( {x,{A}^{ * }z}\right) = 0. \n\]\n\nThis implies the necessity of condition (5.10) for the solvability of (5.9).\n\nConversely, assume that (5.10) is satisfied. In terms of the ort... | Yes |
Theorem 5.7 Let \( A \) be an \( m \times n \) matrix of rank \( r \) with singular system \( \left( {{\mu }_{j},{u}_{j},{v}_{j}}\right) \) and let \( \alpha > 0 \) . Then for each \( y \in {\mathbb{C}}^{m} \) the linear system\n\n\[ \n\alpha {x}_{\alpha } + {A}^{ * }A{x}_{\alpha } = {A}^{ * }y \n\]\n\n(5.18)\n\nis uni... | Proof. For \( \alpha > 0 \) the matrix \( {\alpha I} + {A}^{ * }A \) is positive definite and therefore nonsingular. Since\n\n\[ \n\alpha {u}_{j} + {A}^{ * }A{u}_{j} = \left( {\alpha + {\mu }_{j}^{2}}\right) {u}_{j} \n\]\n\na singular system for the matrix \( {\alpha I} + {A}^{ * }A \) is given by \( \left( {\alpha + {... | Yes |
Corollary 5.8 Under the assumptions of Theorem 5.7 we have convergence:\n\n\[ \mathop{\lim }\limits_{{\alpha \rightarrow 0}}{\left( \alpha I + {A}^{ * }A\right) }^{-1}{A}^{ * }y = {A}^{ \dagger }y. \] | Proof. This is obvious from (5.13) and (5.19). | Yes |
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