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Proposition 35.42. Given a ring homomomorphism \( \rho : A \rightarrow B \), for any two \( A \) -modules \( M \) and \( N \), there is a unique isomorphism\n\n\[ \n{\rho }^{ * }\left( M\right) { \otimes }_{B}{\rho }^{ * }\left( N\right) \approx {\rho }^{ * }\left( {M{ \otimes }_{A}N}\right)\n\]\n\nsuch that \( \left( ... | The proof uses identities from Proposition 33.13. It is not hard but it requires a little gymnastic; a good exercise for the reader. | No |
Proposition 36.1. Let \( f : E \rightarrow E \) and \( {f}^{\prime } : {E}^{\prime } \rightarrow {E}^{\prime } \) be two linear maps over the vector spaces \( E \) and \( {E}^{\prime } \) . A linear map \( g : E \rightarrow {E}^{\prime } \) can be viewed as a linear map between the \( K\left\lbrack X\right\rbrack \) -m... | Proof. First, suppose \( g \) is \( K\left\lbrack X\right\rbrack \) -linear. Then, we have\n\n\[ g\left( {p \cdot {}_{f}u}\right) = p \cdot {}_{{f}^{\prime }}g\left( u\right) \]\n\nfor all \( p \in K\left\lbrack X\right\rbrack \) and all \( u \in E \), so for \( p = X \) we get\n\n\[ g\left( {p{ \cdot }_{f}u}\right) = ... | Yes |
Let \( f : {\mathbb{R}}^{4} \rightarrow {\mathbb{R}}^{4} \) be defined as \( f\left( {x, y, z, w}\right) = \left( {x + w, y + z, y + z, x + w}\right) \). In terms of the standard basis, \( f \) has the matrix representation \( M = \left( \begin{array}{llll} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 0 &... | A basic calculation shows that \( {\chi }_{f}\left( X\right) = {X}^{2}{\left( X - 2\right) }^{2} \) and that \( {m}_{f}\left( X\right) = X\left( {X - 2}\right) \). The primary decomposition theorem implies that \( {\mathbb{R}}^{4} = {W}_{1} \oplus {W}_{2},\;{W}_{1} = \operatorname{Ker}\left( M\right) ,\;{W}_{2} = \oper... | Yes |
Proposition 36.8. Let \( f : E \rightarrow E \) be a linear map on a \( K \) -vector space \( E \), and let \( \left( {{q}_{1},\ldots ,{q}_{n}}\right) \) be the similarity invariants of \( f \) . If \( L \) is a field extension of \( K \) (which means that \( K \subseteq L \) ), and if \( {E}_{\left( L\right) } = L{ \o... | Proof. We know that \( {E}_{f} \) is isomorphic to the direct sum\n\n\[ \n{E}_{f} \approx K\left\lbrack X\right\rbrack /\left( {{q}_{1}K\left\lbrack X\right\rbrack }\right) \oplus \cdots \oplus K\left\lbrack X\right\rbrack /\left( {{q}_{n}K\left\lbrack X\right\rbrack }\right) , \n\]\n\nso by tensoring with \( L\left\lb... | Yes |
Proposition 36.12. For any linear map \( f : E \rightarrow E \) over a \( K \) -vector space \( E \) of dimension \( n \), if \( \left( {{q}_{1},\ldots ,{q}_{n}}\right) \) are the similarity invariants of \( f \), for any matrix \( M \) representing \( f \) with respect to any basis, then for \( k = 1,\ldots, n \) the ... | Note that the matrix \( {XI} - M \) is none other than the matrix that yields the characteristic polynomial \( {\chi }_{f}\left( X\right) = \det \left( {{XI} - M}\right) \) of \( f \) . | No |
Proposition 36.13. For any linear map \( f : E \rightarrow E \) over a \( K \) -vector space \( E \) of dimension \( n \), if \( \left( {{q}_{1},\ldots ,{q}_{n}}\right) \) are the similarity invariants of \( f \), then the following properties hold:\n\n(1) If \( {\chi }_{f}\left( X\right) \) is the characteristic polyn... | Proof. Property (1) follows from Proposition 36.12 for \( k = n \) . It also follows from Theorem 36.6 and the fact that for the companion matrix associated with \( {q}_{i} \), the characteristic polynomial of this matrix is also \( {q}_{i} \) . Property (2) is obvious from (1). Since each \( {q}_{i} \) divides \( {q}_... | Yes |
Theorem 36.17. (Jordan Canonical Form) Let \( E \) be finite-dimensional \( K \) -vector space. The following properties are equivalent:\n\n(1) The eigenvalues of \( f \) all belong to \( K \) .\n\n(2) There is a basis of \( E \) in which the matrix of \( f \) is upper (or lower) triangular.\n\n(3) There exist a basis ... | Proof. The implication (1) \( \Rightarrow \) (3) follows from Theorem 36.15 and Proposition 36.16. The implications \( \left( 3\right) \Rightarrow \left( 2\right) \) and \( \left( 2\right) \Rightarrow \left( 1\right) \) are trivial. | Yes |
Theorem 36.21. If \( M \) is an \( m \times n \) matrix over a PID \( A \), then there exist some invertible \( n \times n \) matrix \( P \) and some invertible \( m \times m \) matrix \( Q \), where \( P \) and \( Q \) are products of elementary matrices and matrices of the form\n\n\[ \left( \begin{matrix} x & y & 0 &... | Proof sketch. In Step 2a, if \( {a}_{11} \) does not divide \( {a}_{k1} \), then first permute row 2 and row \( k \) (if \( k \neq 2) \) . Then, if we write \( a = {a}_{11} \) and \( b = {a}_{k1} \), if \( d \) is a gcd of \( a \) and \( b \) and if \( x, y, s, t \) are determined as explained above, multiply on the le... | Yes |
Let \( E = {\mathbb{R}}^{n} \) (or \( E = {\mathbb{C}}^{n} \) ). There are three standard norms. For every \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in E \), we have the norm \( \parallel x{\parallel }_{1} \), defined such that,\n\n\[ \parallel x{\parallel }_{1} = \left| {x}_{1}\right| + \cdots + \left| {x}_{n}\right... | We proved in Proposition 9.1 that the \( {\ell }_{p} \) -norms are indeed norms. | Yes |
Proposition 37.1. Given a metric space \( E \) with metric \( d \), the family \( \mathcal{O} \) of all open sets defined in Definition 37.4 satisfies the following properties:\n\n(O1) For every finite family \( {\left( {U}_{i}\right) }_{1 \leq i \leq n} \) of sets \( {U}_{i} \in \mathcal{O} \), we have \( {U}_{1} \cap... | Proof. It is straightforward. For the last point, letting \( \rho = d\left( {a, b}\right) /3 \) (in fact \( \rho = d\left( {a, b}\right) /2 \) works too), we can pick \( {U}_{a} = {B}_{0}\left( {a,\rho }\right) \) and \( {U}_{b} = {B}_{0}\left( {b,\rho }\right) \) . By the triangle inequality, we must have \( {U}_{a} \... | Yes |
Proposition 37.2. Let \( \\left( {E, d}\\right) \) be a metric space. For any nonempty subset \( A \) of \( E \) and for any two points \( x, y \\in E \), we have\n\n\[ \n\\left| {d\\left( {x, A}\\right) - d\\left( {y, A}\\right) }\\right| \\leq d\\left( {x, y}\\right) .\n\] | Proof. For all \( a \\in A \) we have\n\n\[ \nd\\left( {x, a}\\right) \\leq d\\left( {x, y}\\right) + d\\left( {y, a}\\right) \n\]\n\nwhich implies\n\n\[ \nd\\left( {x, A}\\right) = \\mathop{\\inf }\\limits_{{a \\in A}}d\\left( {x, a}\\right) \n\]\n\n\[ \n\\leq \\mathop{\\inf }\\limits_{{a \\in A}}\\left( {d\\left( {x,... | Yes |
Proposition 37.3. Let \( \\left( {E, d}\\right) \) be a metric space. For any nonempty subset \( A \) of \( E \) , and any \( r > 0 \), the set \( {V}_{r}\\left( A\\right) \) is an open set containing \( A \) . | Proof. For any \( y \\in E \) such that \( d\\left( {x, y}\\right) < r - d\\left( {x, A}\\right) \), by Proposition 37.2 we have\n\n\[ d\\left( {y, A}\\right) \\leq d\\left( {x, A}\\right) + d\\left( {x, y}\\right) \\leq d\\left( {x, A}\\right) + r - d\\left( {x, A}\\right) = r, \]\n\nso \( {V}_{r}\\left( A\\right) \) ... | Yes |
Proposition 37.4. Given a topological space \( \left( {E,\mathcal{O}}\right) \), given any subset \( A \) of \( E \), the closure \( \bar{A} \) of \( A \) is the set of all points \( x \in E \) such that for every open set \( U \) containing \( x \), then \( U \cap A \neq \varnothing \) . | Proof. If \( A = \varnothing \), since \( \varnothing \) is closed, the proposition holds trivially. Thus, assume that \( A \neq \varnothing \) . First assume that \( x \in \bar{A} \) . Let \( U \) be any open set such that \( x \in U \) . If \( U \cap A = \varnothing \), since \( U \) is open, then \( E - U \) is a cl... | Yes |
Proposition 37.5. Given a topological space \( \left( {E,\mathcal{O}}\right) \), given any subset \( A \) of \( E \), let\n\n\[ \mathcal{U} = \{ U \cap A \mid U \in \mathcal{O}\} \]\n\nbe the family of all subsets of \( A \) obtained as the intersection of any open set in \( \mathcal{O} \) with \( A \) . The following ... | Proof. Left as an exercise. | No |
Proposition 37.6. Given a topological space \( \left( {E,\mathcal{O}}\right) \), given any subset \( A \) of \( E \), if \( \mathcal{U} \) is the subspace topology, then the following properties hold.\n\n(1) If \( A \) is an open set \( A \in \mathcal{O} \), then every open set \( U \in \mathcal{U} \) is an open set \(... | Proof. Left as an exercise. | No |
Proposition 37.7. Given \( n \) topological spaces \( \left( {{E}_{i},{\mathcal{O}}_{i}}\right) \), let \( \mathcal{B} \) be the family of subsets of \( {E}_{1} \times \cdots \times {E}_{n} \) defined as follows:\n\n\[ \mathcal{B} = \left\{ {{U}_{1} \times \cdots \times {U}_{n} \mid {U}_{i} \in {\mathcal{O}}_{i},1 \leq... | Proof. Left as an exercise. | No |
Proposition 37.9. Let \( \left( {E,{\mathcal{O}}_{E}}\right) \) and \( \left( {F,{\mathcal{O}}_{F}}\right) \) be topological spaces, and let \( f : E \rightarrow F \) be a function. For every \( a \in E \), the function \( f \) is continuous at \( a \in E \) iff for every neighborhood \( N \) of \( f\left( a\right) \in... | If \( E \) and \( F \) are metric spaces defined by metrics \( {d}_{E} \) and \( {d}_{F} \), we can show easily that \( f \) is continuous at \( a \) iff\n\nfor every \( \epsilon > 0 \), there is some \( \eta > 0 \), such that, for every \( x \in E \) ,\n\n\[ \text{if}{d}_{E}\left( {a, x}\right) \leq \eta \text{, then}... | No |
Proposition 37.11. If \( E \) is a topological space, and \( \left( {\mathbb{R},\left| {x - y}\right| }\right) \) the reals under the standard topology, for any two functions \( f : E \rightarrow \mathbb{R} \) and \( g : E \rightarrow \mathbb{R} \), for any \( a \in E \), for any \( \lambda \in \mathbb{R} \), if \( f \... | Proof. Left as an exercise. | No |
Proposition 37.12. Given a topological space \( \left( {E,\mathcal{O}}\right) \), if the Hausdorff separation axiom holds, then every sequence has at most one limit. | Proof. Left as an exercise. | No |
Proposition 37.13. Given any metric space \( \left( {E, d}\right) \), for any subset \( A \) of \( E \) and any point \( x \in E \), we have \( x \in \bar{A} \) iff there is a sequence \( \left( {a}_{n}\right) \) of points \( {a}_{n} \in A \) converging to \( x \) . | Proof. If the sequence \( \left( {a}_{n}\right) \) of points \( {a}_{n} \in A \) converges to \( x \), then for every open subset \( U \) of \( E \) containing \( x \), there is some \( {n}_{0} \) such that \( {a}_{n} \in U \) for all \( n \geq {n}_{0} \), so \( U \cap A \neq \varnothing \), and Proposition 37.4 implie... | Yes |
Proposition 37.16. A subset of the real line, \( \mathbb{R} \), is connected iff it is an interval, i.e., of the form \( \left\lbrack {a, b}\right\rbrack ,(a, b\rbrack \), where \( a = - \infty \) is possible, \( \lbrack a, b) \), where \( b = + \infty \) is possible, or \( \left( {a, b}\right) \) , where \( a = - \inf... | Proof. Assume that \( A \) is a connected nonempty subset of \( \mathbb{R} \) . The cases where \( A = \varnothing \) or \( A \) consists of a single point are trivial. Otherwise, we show that whenever \( a, b \in A, a < b \), then the entire interval \( \left\lbrack {a, b}\right\rbrack \) is a subset of \( A \) . Inde... | Yes |
Proposition 37.17. A topological space is connected iff every locally constant function is constant. | Proof. First, assume that \( X \) is connected. Let \( f : X \rightarrow Y \) be a locally constant function to some space \( Y \) and assume that \( f \) is not constant. Pick any \( y \in f\left( X\right) \) . Since \( f \) is not constant, \( {U}_{1} = {f}^{-1}\left( y\right) \neq X \), and of course, \( {U}_{1} \ne... | Yes |
Proposition 37.18. Given any continuous map, \( f : E \rightarrow F \), if \( A \subseteq E \) is connected, then \( f\left( A\right) \) is connected. | Proof. If \( f\left( A\right) \) is not connected, then there exist some nonempty open sets, \( U, V \), in \( F \) such that \( f\left( A\right) \cap U \) and \( f\left( A\right) \cap V \) are nonempty and disjoint, and\n\n\[ f\left( A\right) = \left( {f\left( A\right) \cap U}\right) \cup \left( {f\left( A\right) \cap... | Yes |
Lemma 37.19. Given a topological space, \( E \), for any family, \( {\left( {A}_{i}\right) }_{i \in I} \), of (nonempty) connected subsets of \( E \), if \( {A}_{i} \cap {A}_{j} \neq \varnothing \) for all \( i, j \in I \), then the union, \( A = \mathop{\bigcup }\limits_{{i \in I}}{A}_{i} \), of the family, \( {\left(... | Proof. Assume that \( \mathop{\bigcup }\limits_{{i \in I}}{A}_{i} \) is not connected. There exists two nonempty open subsets, \( U \) and \( V \), of \( E \) such that \( A \cap U \) and \( A \cap V \) are disjoint and nonempty and such that\n\n\[ A = \left( {A \cap U}\right) \cup \left( {A \cap V}\right) \]\n\nNow, f... | Yes |
Lemma 37.20. If \( A \) is a connected subset of a topological space, E, then for every subset, \( B \), such that \( A \subseteq B \subseteq \bar{A} \), where \( \bar{A} \) is the closure of \( A \) in \( E \), the set \( B \) is connected. | Proof. If \( B \) is not connected, then there are two nonempty open subsets, \( U, V \), of \( E \) such that \( B \cap U \) and \( B \cap V \) are disjoint and nonempty, and\n\n\[ B = \left( {B \cap U}\right) \cup \left( {B \cap V}\right) .\n\]\n\nSince \( A \subseteq B \), the above implies that\n\n\[ A = \left( {A ... | Yes |
Proposition 37.22. A topological space, \( E \), is locally connected iff for every open subset, \( A \), of \( E \), the connected components of \( A \) are open. | Proof. Assume that \( E \) is locally connected. Let \( A \) be any open subset of \( E \) and let \( C \) be one of the connected components of \( A \) . For any \( a \in C \subseteq A \), there is some connected neighborhood, \( U \), of \( a \) such that \( U \subseteq A \) and since \( C \) is a connected component... | Yes |
If a topological space, \( E \), is arcwise connected, then it is connected. | First, assume that \( E \) is arcwise connected. Pick any point, \( a \), in \( E \) . Since \( E \) is arcwise connected, for every \( b \in E \), there is a path, \( {\gamma }_{b} : \left\lbrack {0,1}\right\rbrack \rightarrow E \), from \( a \) to \( b \) and so,\n\n\[ E = \mathop{\bigcup }\limits_{{b \in E}}{\gamma ... | Yes |
Proposition 37.24. A topological Hausdorff space \( E \) is compact iff for every family \( {\left( {F}_{i}\right) }_{i \in I} \) of closed sets having the finite intersection property, then \( \mathop{\bigcap }\limits_{{i \in I}}{F}_{i} \neq \varnothing \) . | Proof. If \( E \) is compact and \( {\left( {F}_{i}\right) }_{i \in I} \) is a family of closed sets having the finite intersection property, then \( \mathop{\bigcap }\limits_{{i \in I}}{F}_{i} \) cannot be empty, since otherwise we would have \( \mathop{\bigcap }\limits_{{j \in J}}{F}_{j} = \varnothing \) for some fin... | No |
Proposition 37.25. Every closed interval, \( \left\lbrack {a, b}\right\rbrack \), of the real line is compact. | Proof. We proceed by contradiction. Let \( {\left( {U}_{i}\right) }_{i \in I} \) be any open cover of \( \left\lbrack {a, b}\right\rbrack \) and assume that there is no finite open subcover. Let \( c = \left( {a + b}\right) /2 \) . If both \( \left\lbrack {a, c}\right\rbrack \) and \( \left\lbrack {c, b}\right\rbrack \... | Yes |
Proposition 37.26. Given a topological Hausdorff space, E, for every compact subset, A, and every point, \( b \), not in \( A \), there exist disjoint open sets, \( U \) and \( V \), such that \( A \subseteq U \) and \( b \in V \) . See Figure 37.30. As a consequence, every compact subset is closed. | Proof. Since \( E \) is Hausdorff, for every \( a \in A \), there are some disjoint open sets, \( {U}_{a} \) and \( {V}_{a} \) , containing \( a \) and \( b \) respectively. Thus, the family, \( {\left( {U}_{a}\right) }_{a \in A} \), forms an open cover of \( A \) . Since \( A \) is compact there is a finite open subco... | Yes |
Proposition 37.28. Given a compact topological space, E, every closed set is compact. | Proof. Since \( A \) is closed, \( E - A \) is open and from any open cover, \( {\left( {U}_{i}\right) }_{i \in I} \), of \( A \), we can form an open cover of \( E \) by adding \( E - A \) to \( {\left( {U}_{i}\right) }_{i \in I} \) and, since \( E \) is compact, a finite subcover, \( {\left( {U}_{j}\right) }_{j \in J... | Yes |
Proposition 37.29. Every metrizable space \( E \) is normal. | Proof. Assume the topology of \( E \) is given by the metric \( d \) . Since \( B \) is closed and \( A \cap B = \varnothing \) , for every \( a \in A \) since \( a \notin \bar{B} = B \), there is some open ball \( {B}_{0}\left( {a,{\epsilon }_{a}}\right) \) of radius \( {\epsilon }_{a} > 0 \) such that \( {B}_{0}\left... | Yes |
Proposition 37.30. Given a compact topological space, \( E \), for every \( a \in E \), for every neighborhood, \( V \), of a, there exists a compact neighborhood, \( U \), of a such that \( U \subseteq V \) . | Proof. Since \( V \) is a neighborhood of \( a \), there is some open subset, \( O \), of \( V \) containing \( a \) . Then the complement, \( K = E - O \), of \( O \) is closed and since \( E \) is compact, by Proposition 37.28, \( K \) is compact. Now, if we consider the family of all closed sets of the form, \( K \c... | Yes |
Proposition 37.31. Let \( E \) be a topological space and let \( F \) be a topological Hausdorff space. For every compact subset, \( A \), of \( E \), for every continuous map, \( f : E \rightarrow F \), the subspace \( f\left( A\right) \) is compact. | Proof. Let \( {\left( {U}_{i}\right) }_{i \in I} \) be an open cover of \( f\left( A\right) \) . We claim that \( {\left( {f}^{-1}\left( {U}_{i}\right) \right) }_{i \in I} \) is an open cover of \( A \), which is easily checked. Since \( A \) is compact, there is a finite open subcover, \( {\left( {f}^{-1}\left( {U}_{j... | Yes |
Proposition 37.32. If \( E \) is a compact nonempty topological space and if \( f : E \rightarrow \mathbb{R} \) is a continuous function, then there are points \( a, b \in E \) such that \( f\left( a\right) \) is the minimum of \( f\left( E\right) \) and \( f\left( b\right) \) is the maximum of \( f\left( E\right) \) . | Proof. The set \( f\left( E\right) \) is a compact subset of \( \mathbb{R} \) and thus, a closed and bounded set which contains its greatest lower bound and its least upper bound. | Yes |
Proposition 37.33. Let \( \\left( {E, d}\\right) \) be a metric space. For any nonempty subset \( A \) of \( E \), if \( A \) is compact, then for every open subset \( U \) such that \( A \\subseteq U \), there is some \( r > 0 \) such that \( {V}_{r}\\left( A\\right) \\subseteq U \) . | Proof. The function \( x \\mapsto d\\left( {x, E - U}\\right) \) is continuous and \( d\\left( {x, E - U}\\right) > 0 \) for \( x \\in A \) (since \( A \\subseteq U) \) . By Proposition 37.32, there is some \( a \\in A \) such that\n\n\[ d\\left( {a, E - U}\\right) = \\mathop{\\inf }\\limits_{{x \\in A}}d\\left( {x, E ... | Yes |
Proposition 37.34. Given a locally compact topological space, \( E \), for every \( a \in E \), for every neighborhood, \( N \), of a, there exists a compact neighborhood, \( U \), of a, such that \( U \subseteq N \) . | Proof. For any \( a \in E \), there is some compact neighborhood, \( V \), of \( a \) . By Proposition 37.30, every neigborhood of \( a \) relative to \( V \) contains some compact neighborhood \( U \) of \( a \) relative to \( V \) . But every neighborhood of \( a \) relative to \( V \) is a neighborhood of \( a \) re... | Yes |
Proposition 37.35. Let \( \\left( {E, d}\\right) \) be a metric space. If \( E \) is locally compact, then for any nonempty compact subset \( A \) of \( E \), there is some \( r > 0 \) such that \( \\overline{{V}_{r}\\left( A\\right) } \) is compact. | Proof. Since \( E \) is locally compact, for every \( x \\in A \), there is some compact subset \( {V}_{x} \) whose interior \( {V}_{x} \) contains \( x \) . The family of open subsets \( {V}_{x} \) is an open cover \( A \), and since \( A \) is compact, it has a finite subcover \( \\left\\{ {{V}_{{x}_{1}},\\ldots ,{V}... | Yes |
Theorem 37.36. Let \( E \) be a locally compact topological space. The Alexandroff compactification, \( {E}_{\omega } \), of \( E \) is a compact space such that \( E \) is a subspace of \( {E}_{\omega } \) and if \( E \) is not compact, then \( \bar{E} = {E}_{\omega } \) . | Proof. The verification that \( {\mathcal{O}}_{\omega } \) is a family of open sets is not difficult but a bit tedious. Details can be found in Munkres [129] or Schwartz [148]. Let us show that \( {E}_{\omega } \) is compact. For every open cover, \( {\left( {U}_{i}\right) }_{i \in I} \), of \( {\bar{E}}_{\omega } \), ... | Yes |
Proposition 37.37. If \( E \) is a metric space, then \( E \) is second-countable iff \( E \) is separable. | Proof. If \( \mathcal{B} = \left( {B}_{n}\right) \) is a countable basis for the topology of \( E \), then for any set \( S \) obtained by picking some point \( {s}_{n} \) in \( {B}_{n} \), since every nonempty open subset \( U \) of \( E \) is the union of some of the \( {B}_{n} \), the intersection \( U \cap S \) is ... | Yes |
Proposition 37.38. If \( E \) is a compact metric space, then \( E \) is separable. | Proof. For every \( n > 0 \), the family of open balls of radius \( 1/n \) forms an open cover of \( E \) , and since \( E \) is compact, there is a finite subset \( {A}_{n} \) of \( E \) such that \( E = \mathop{\bigcup }\limits_{{{a}_{i} \in {A}_{n}}}{B}_{0}\left( {{a}_{i},1/n}\right) \) . It is easy to see that this... | Yes |
Theorem 37.39. (Urysohn metrization theorem) If a topological space \( E \) is regular and second-countable, then it is metrizable. | The proof of Theorem 37.39 can be found in Munkres [129] (Chapter 4, Theorem 34.1). | No |
Proposition 37.40. Let \( E \) be a locally compact metric space. The following properties are equivalent:\n\n(1) There is a sequence \( {\left( {U}_{n}\right) }_{n \geq 0} \) of open subsets such that for all \( n \in \mathbb{N},{U}_{n} \subseteq {U}_{n + 1},\overline{{U}_{n}} \) is compact, \( \overline{{U}_{n}} \sub... | Proof. We show (1) implies (2), (2) implies (3), and (3) implies (1). Obviously, (1) implies (2) since the \( \overline{{U}_{n}} \) are compact.\n\nIf (2) holds, then \( E = \mathop{\bigcup }\limits_{{n \geq 0}}{K}_{n} \), for some compact subsets \( {K}_{n} \) . By Proposition 37.38, each compact subset \( {K}_{n} \) ... | Yes |
Proposition 37.41. Given a second-countable topological space \( E \), every open cover \( {\left( {U}_{i}\right) }_{i \in I} \) , of \( E \) contains some countable subcover. | Proof. Let \( {\left( {O}_{n}\right) }_{n \geq 0} \) be a countable basis for the topology. Then all sets \( {O}_{n} \) contained in some \( {U}_{i} \) can be arranged into a countable subsequence, \( {\left( {\Omega }_{m}\right) }_{m \geq 0} \), of \( {\left( {O}_{n}\right) }_{n \geq 0} \) and for every \( {\Omega }_{... | Yes |
Proposition 37.42. Given a second-countable topological Hausdorff space, E, a point, l, is an accumulation point of the sequence, \( \left( {x}_{n}\right) \), iff \( l \) is the limit of some subsequence, \( \left( {x}_{{n}_{k}}\right) \), of \( \left( {x}_{n}\right) \) . | Proof. Clearly, if \( l \) is the limit of some subsequence \( \left( {x}_{{n}_{k}}\right) \) of \( \left( {x}_{n}\right) \), it is an accumulation point of \( \left( {x}_{n}\right) \).\n\nConversely, let \( {\left( {U}_{k}\right) }_{k \geq 0} \) be the sequence of open sets containing \( l \), where each \( {U}_{k} \)... | Yes |
Lemma 37.44. Given a metric space, \( E \), if every sequence, \( \left( {x}_{n}\right) \), has an accumulation point, for every open cover, \( {\left( {U}_{i}\right) }_{i \in I} \), of \( E \), there is some \( \delta > 0 \) (a Lebesgue number for \( {\left( {U}_{i}\right) }_{i \in I} \) ) such that, for every open ba... | Proof. If there was no \( \delta \) with the above property, then, for every natural number, \( n \), there would be some open ball, \( {B}_{0}\left( {{a}_{n},1/n}\right) \), which is not contained in any open set, \( {U}_{i} \), of the open cover, \( {\left( {U}_{i}\right) }_{i \in I} \) . However, the sequence, \( \l... | Yes |
Theorem 37.45. Given two metric spaces, \( \left( {E,{d}_{E}}\right) \) and \( \left( {F,{d}_{F}}\right) \), if \( E \) is compact and if \( f : E \rightarrow F \) is a continuous function, then \( f \) is uniformly continuous. | Proof. Consider any \( \epsilon > 0 \) and let \( {\left( {B}_{0}\left( y,\epsilon /2\right) \right) }_{y \in F} \) be the open cover of \( F \) consisting of open balls of radius \( \epsilon /2 \) . Since \( f \) is continuous, the family,\n\n\[{\left( {f}^{-1}\left( {B}_{0}\left( y,\epsilon /2\right) \right) \right) ... | Yes |
Lemma 37.46. Given a metric space, \( E \), if every sequence, \( \left( {x}_{n}\right) \), has an accumulation point, then for every \( \epsilon > 0 \), there is a finite open cover, \( {B}_{0}\left( {{a}_{0},\epsilon }\right) \cup \cdots \cup {B}_{0}\left( {{a}_{n},\epsilon }\right) \), of \( E \) by open balls of ra... | Proof. Let \( {a}_{0} \) be any point in \( E \) . If \( {B}_{0}\left( {{a}_{0},\epsilon }\right) = E \), then the lemma is proved. Otherwise, assume that a sequence, \( \left( {{a}_{0},{a}_{1},\ldots ,{a}_{n}}\right) \), has been defined, such that \( {B}_{0}\left( {{a}_{0},\epsilon }\right) \cup \cdots \cup {B}_{0}\l... | Yes |
Theorem 37.47. A metric space, \( E \), is compact iff every sequence, \( \left( {x}_{n}\right) \), has an accumulation point. | Proof. We already observed that the proof of Proposition 37.43 shows that for any compact space (not necessarily metric), every sequence, \( \left( {x}_{n}\right) \), has an accumulation point. Conversely, let \( E \) be a metric space, and assume that every sequence, \( \left( {x}_{n}\right) \), has an accumulation po... | Yes |
Proposition 37.48. Given a metric space, \( E \), if a Cauchy sequence, \( \left( {x}_{n}\right) \), has some accumulation point, \( a \), then \( a \) is the limit of the sequence, \( \left( {x}_{n}\right) \) . | Proof. Since \( \left( {x}_{n}\right) \) is a Cauchy sequence, for every \( \epsilon > 0 \), there is some \( p \geq 0 \), such that, for all \( m, n \geq p \), then \( d\left( {{x}_{m},{x}_{n}}\right) \leq \epsilon /2 \) . Since \( a \) is an accumulation point for \( \left( {x}_{n}\right) \), for infinitely many \( n... | Yes |
Proposition 37.50. Let \( \left( {E, d}\right) \) be a metric space, and let \( A \) be a subset of \( E \) . If \( A \) is complete (which means that every Cauchy sequence of elements in \( A \) converges to some point of \( A \) ), then \( A \) is closed in \( E \) . | Proof. Assume \( x \in \bar{A} \) . By Proposition 37.13, there is some sequence \( \left( {a}_{n}\right) \) of points \( {a}_{n} \in A \) which converges to \( x \) . Consequently \( \left( {a}_{n}\right) \) is a Cauchy sequence in \( E \), and thus a Cauchy sequence in \( A \) (since \( {a}_{n} \in A \) for all \( n ... | Yes |
Proposition 37.51. Let \( \\left( {E, d}\\right) \) be a metric space, and let \( A \) be a subset of \( E \) . If \( E \) is complete and if \( A \) is closed in \( E \), then \( A \) is complete. | Proof. Let \( \\left( {a}_{n}\\right) \) be a Cauchy sequence in \( A \) . The sequence \( \\left( {a}_{n}\\right) \) is also a Cauchy sequence in \( E \), and since \( E \) is complete, it has a limit \( x \\in E \) . But \( {a}_{n} \\in A \) for all \( n \), so by Proposition 37.13 we must have \( x \\in \\bar{A} \) ... | Yes |
Theorem 37.53. Let \( \left( {E, d}\right) \) be any metric space. There is a complete metric space \( \left( {\widehat{E},\widehat{d}}\right) \) called a completion of \( \left( {E, d}\right) \), and a distance-preserving (uniformly continuous) map \( \varphi : E \rightarrow \) \( \widehat{E} \) such that \( \varphi \... | Proof. Consider the set \( \mathcal{E} \) of all Cauchy sequences \( \left( {x}_{n}\right) \) in \( E \), and define the relation \( \sim \) on \( \mathcal{E} \) as follows:\n\n\[ \left( {x}_{n}\right) \sim \left( {y}_{n}\right) \;\text{ iff }\;\mathop{\lim }\limits_{{n \mapsto \infty }}d\left( {{x}_{n},{y}_{n}}\right)... | Yes |
Proposition 37.54. If \( \left( {E, d}\right) \) is a nonempty complete metric space, every contraction mapping, \( f : E \rightarrow E \), has a unique fixed point. Furthermore, for every \( {x}_{0} \in E \), defining the sequence, \( \left( {x}_{n}\right) \), such that \( {x}_{n + 1} = f\left( {x}_{n}\right) \), the ... | Proof. First we prove that \( f \) has at most one fixed point. Indeed, if \( f\left( a\right) = a \) and \( f\left( b\right) = b \) , since\n\n\[ d\left( {a, b}\right) = d\left( {f\left( a\right), f\left( b\right) }\right) \leq {kd}\left( {a, b}\right) \]\n\nand \( 0 \leq k < 1 \), we must have \( d\left( {a, b}\right... | Yes |
Theorem 37.55. If \( \left( {X, d}\right) \) is a metric space, then the Hausdorff distance, \( D \), on the set, \( \mathcal{K}\left( X\right) \), of nonempty compact subsets of \( X \) is a distance. If \( \left( {X, d}\right) \) is complete, then \( \left( {\mathcal{K}\left( X\right), D}\right) \) is complete and if... | Proof. Since (nonempty) compact sets are bounded, \( D\left( {A, B}\right) \) is well defined. Clearly \( D \) is symmetric. Assume that \( D\left( {A, B}\right) = 0 \) . Then for every \( \epsilon > 0, A \subseteq {V}_{\epsilon }\left( B\right) \), which means that for every \( a \in A \), there is some \( b \in B \) ... | Yes |
Proposition 37.56. Given two normed vector spaces \( E \) and \( F \), for any linear map \( f : E \rightarrow \) \( F \), the following conditions are equivalent:\n\n(1) The function \( f \) is continuous at 0 .\n\n(2) There is a constant \( k \geq 0 \) such that,\n\n\[ \parallel f\left( u\right) \parallel \leq k\text... | Proof. Assume (1). Then for every \( \epsilon > 0 \), there is some \( \eta > 0 \) such that, for every \( u \in E \), if \( \parallel u\parallel \leq \eta \), then \( \parallel f\left( u\right) \parallel \leq \epsilon \) . Pick \( \epsilon = 1 \), so that there is some \( \eta > 0 \) such that, if \( \parallel u\paral... | Yes |
Proposition 37.57. If \( E = {\mathbb{R}}^{n} \) or \( E = {\mathbb{C}}^{n} \), with any of the norms \( {\begin{Vmatrix}{\parallel }_{1},\parallel \end{Vmatrix}}_{2} \), or \( \parallel {\parallel }_{\infty } \), and \( F \) is any normed vector space, then every linear map \( f : E \rightarrow F \) is continuous. | Proof. Let \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) be the standard basis of \( {\mathbb{R}}^{n} \) (a similar proof applies to \( {\mathbb{C}}^{n} \) ). In view of Proposition 9.3, it is enough to prove the proposition for the norm\n\n\[ \parallel x{\parallel }_{\infty } = \max \left\{ {\left| {x}_{i}\right| \mid... | Yes |
Proposition 37.59. Given normed vector spaces \( E, F \) and \( G \), for any bilinear map \( f : E \times E \rightarrow G \), the following conditions are equivalent:\n\n(1) The function \( f \) is continuous at \( \langle 0,0\rangle \) .\n\n2) There is a constant \( k \geq 0 \) such that,\n\n\[ \parallel f\left( {u, ... | Proof. It is similar to that of Proposition 37.56, with a small subtlety in proving that (3) implies (4), namely that two different \( \eta \) ’s that are not independent are needed. | No |
Theorem 37.63. If \( \\left( {E,\\parallel \\parallel }\\right) \) is a normed vector space, then its completion \( \\left( {\\widehat{E},\\widehat{d}}\\right) \) as a metric space (where \( E \) is given the metric \( d\\left( {x, y}\\right) = \\parallel x - y\\parallel \) ) can be given a unique vector space structur... | Proof. The addition operation \( + : E \\times E \\rightarrow E \) is uniformly continuous because\n\n\[ \n\\begin{Vmatrix}{\\left( {{u}^{\\prime } + {v}^{\\prime }}\\right) - \\left( {{u}^{\\prime \\prime } + {v}^{\\prime \\prime }}\\right) }\\end{Vmatrix} \\leq \\begin{Vmatrix}{{u}^{\\prime } - {u}^{\\prime \\prime }... | Yes |
Theorem 38.1. If \( \left( {X, d}\right) \) is a nonempty complete metric space, then every iterated function system, \( \left( {{f}_{1},\ldots ,{f}_{n}}\right) \), has a unique invariant set, \( A \), which is a nonempty compact subset of \( X \). Furthermore, for every nonempty compact subset, \( {A}_{0} \), of \( X ... | Proof. Since \( X \) is complete, by Theorem 37.55, the space \( \left( {\mathcal{K}\left( X\right), D}\right) \) is a complete metric space. The theorem will follow from Theorem 37.54 if we can show that the map, \( F : \mathcal{K}\left( X\right) \rightarrow \mathcal{K}\left( X\right) \), defined such that\n\n\[ F\lef... | Yes |
Consider an equilateral triangle with vertices \( a, b, c \), and let \( {f}_{1},{f}_{2},{f}_{3} \) be the dilatations of centers \( a, b, c \) and ratio \( 1/2 \) . The Sierpinski gasket is the invariant set of the ifs \( \left( {{f}_{1},{f}_{2},{f}_{3}}\right) \) . The dilations \( {f}_{1},{f}_{2},{f}_{3} \) can be d... | \[ \begin{align*} {x}^{\prime } &= \frac{1}{2}x - \frac{1}{4} \\ {y}^{\prime } &= \frac{1}{2}y \\ {x}^{\prime } &= \frac{1}{2}x + \frac{1}{4} \\ {y}^{\prime } &= \frac{1}{2}y \\ {x}^{\prime } &= \frac{1}{2}x \\ {y}^{\prime } &= \frac{1}{2}y + \frac{\sqrt{3}}{4} \end{align*} \] | Yes |
The Sierpinski dragon is specified by the following three contractions: | \[ {x}^{\prime } = - \frac{1}{4}x - \frac{\sqrt{3}}{4}y + \frac{3}{4} \]\n\[ {y}^{\prime } = \frac{\sqrt{3}}{4}x - \frac{1}{4}y + \frac{\sqrt{3}}{4} \]\n\[ {x}^{\prime } = - \frac{1}{4}x + \frac{\sqrt{3}}{4}y - \frac{3}{4} \]\n\[ {y}^{\prime } = - \frac{\sqrt{3}}{4}x - \frac{1}{4}y + \frac{\sqrt{3}}{4} \]\n\[ {x}^{\pri... | Yes |
The Heighway dragon is specified by the following two contractions:\n\n\[ \n{x}^{\prime } = \frac{1}{2}x - \frac{1}{2}y \]\n\n\[ \n{y}^{\prime } = \frac{1}{2}x + \frac{1}{2}y \]\n\n\[ \n{x}^{\prime } = - \frac{1}{2}x - \frac{1}{2}y \]\n\n\[ \n{y}^{\prime } = \frac{1}{2}x - \frac{1}{2}y + 1 \]\n\nIt can be shown that fo... | The result of 13 iterations, starting with the line segment \( \left( {\left( {0,0}\right) ,\left( {0,1}\right) }\right) \), is shown in Figure 38.4. | No |
The Koch curve is specified by the following four contractions: | \[ \n{x}^{\prime } = \frac{1}{3}x - \frac{2}{3} \]\n\[ \n{y}^{\prime } = \frac{1}{3}y \]\n\[ \n{x}^{\prime } = \frac{1}{6}x - \frac{\sqrt{3}}{6}y - \frac{1}{6} \]\n\[ \n{y}^{\prime } = \frac{\sqrt{3}}{6}x + \frac{1}{6}y + \frac{\sqrt{3}}{6} \]\n\[ \n{x}^{\prime } = \frac{1}{6}x + \frac{\sqrt{3}}{6}y + \frac{1}{6} \]\n\... | Yes |
This version of the Hilbert curve is defined by the following four contractions: | \[ {x}^{\prime } = \frac{1}{2}x - \frac{1}{2} \] \[ {y}^{\prime } = \frac{1}{2}y + 1 \] \[ {x}^{\prime } = \frac{1}{2}x + \frac{1}{2} \] \[ {y}^{\prime } = \frac{1}{2}y + 1 \] \[ {x}^{\prime } = - \frac{1}{2}y + 1 \] \[ {y}^{\prime } = \frac{1}{2}x + \frac{1}{2} \] \[ {x}^{\prime } = \frac{1}{2}y - 1 \] \[ {y}^{\prime ... | Yes |
Example 39.1. Let \( f : {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) be the function given by\n\n\[ f\left( {x, y}\right) = \left\{ \begin{array}{ll} \frac{{x}^{2}y}{{x}^{4} + {y}^{2}} & \text{ if }\left( {x, y}\right) \neq \left( {0,0}\right) \\ 0 & \text{ if }\left( {x, y}\right) = \left( {0,0}\right) \end{array}\righ... | For any \( u \neq 0 \), letting \( u = \left( \begin{array}{l} h \\ k \end{array}\right) \), we have\n\n\[ \frac{f\left( {0 + {tu}}\right) - f\left( 0\right) }{t} = \frac{{h}^{2}k}{{t}^{2}{h}^{4} + {k}^{2}} \]\n\nso that\n\n\[ {\mathrm{D}}_{u}f\left( {0,0}\right) = \left\{ \begin{array}{ll} \frac{{h}^{2}}{k} & \text{ i... | Yes |
Proposition 39.1. Let \( E \) and \( F \) be two normed affine spaces, let \( A \) be a nonempty open subset of \( E \), and let \( f : A \rightarrow F \) be any function. For any \( a \in A \), if \( \mathrm{D}f\left( a\right) \) is defined, then \( f \) is continuous at a and \( f \) has a directional derivative \( {... | Proof. If \( L = \mathrm{D}f\left( a\right) \) exists, then for any nonzero vector \( u \in \overrightarrow{E} \), because \( A \) is open, for any \( t \in \mathbb{R} - \{ 0\} \) (or \( t \in \mathbb{C} - \{ 0\} \) ) small enough, \( a + {tu} \in A \), so \[ f\left( {a + {tu}}\right) = f\left( a\right) + L\left( {tu}\... | Yes |
Proposition 39.2. Let \( E \) and \( F \) be two normed affine spaces, let \( A \) be a nonempty open subset of \( E \), and let \( f : A \rightarrow F \) be any function. For any \( a \in A \), if \( \mathrm{D}{f}_{a} \) is defined, then \( f \) is continuous at a iff \( \mathrm{D}{f}_{a} \) is a continuous linear map... | Proof. Proposition 39.1 shows that if \( \mathrm{D}{f}_{a} \) is defined and continuous then \( f \) is continuous at \( a \) . Conversely, assume that \( \mathrm{D}{f}_{a} \) exists and that \( f \) is continuous at \( a \) . Since \( f \) is continuous at \( a \) and since \( \mathrm{D}{f}_{a} \) exists, for any \( \... | Yes |
Proposition 39.3. Given two normed affine spaces \( E \) and \( F \), if \( f : E \rightarrow F \) is a constant function, then \( \mathrm{D}f\left( a\right) = 0 \), for every \( a \in E \) . If \( f : E \rightarrow F \) is a continuous affine map, then \( \mathrm{D}f\left( a\right) = \overrightarrow{f} \), for every \... | Proof. Straightforward. | No |
Proposition 39.4. Given a normed affine space \( E \) and a normed vector space \( F \), for any two functions \( f, g : E \rightarrow F \), for every \( a \in E \), if \( \mathrm{D}f\left( a\right) \) and \( \mathrm{D}g\left( a\right) \) exist, then \( \mathrm{D}\left( {f + g}\right) \left( a\right) \) and \( \mathrm{... | Proof. Straightforward. | No |
Proposition 39.5. Given three normed vector spaces \( {E}_{1},{E}_{2} \), and \( F \), for any continuous bilinear map \( f : {E}_{1} \times {E}_{2} \rightarrow F \), for every \( \left( {a, b}\right) \in {E}_{1} \times {E}_{2},\mathrm{D}f\left( {a, b}\right) \) exists, and for every \( u \in {E}_{1} \) and \( v \in {E... | Proof. Since \( f \) is bilinear, a simple computation implies that\n\n\[ f\left( {\left( {a, b}\right) + \left( {u, v}\right) }\right) - f\left( {a, b}\right) - \left( {f\left( {u, b}\right) + f\left( {a, v}\right) }\right) = f\left( {a + u, b + v}\right) - f\left( {a, b}\right) - f\left( {u, b}\right) - f\left( {a, v... | Yes |
Proposition 39.8. Given two normed affine spaces \( E \) and \( F \) , let \( A \) be some open subset in \( E \), let \( B \) be some open subset in \( F \), let \( f : A \rightarrow B \) be a bijection from \( A \) to \( B \), and assume that \( \mathrm{D}f \) exists on \( A \) and that \( \mathrm{D}{f}^{-1} \) exist... | \[ \mathrm{D}{f}^{-1}\left( {f\left( a\right) }\right) = {\left( \mathrm{D}f\left( a\right) \right) }^{-1}. \] | No |
Proposition 39.9. Given normed affine spaces \( E \) and \( F = \left( {{F}_{1},{b}_{1}}\right) \oplus \cdots \oplus \left( {{F}_{m},{b}_{m}}\right) \), given any open subset \( A \) of \( E \), for any \( a \in A \), for any function \( f : A \rightarrow F \), letting \( f = \left( {{f}_{1},\ldots ,{f}_{m}}\right) \) ... | Proof. Observe that \( f\left( {a + h}\right) - f\left( a\right) = \left( {f\left( {a + h}\right) - b}\right) - \left( {f\left( a\right) - b}\right) \), where \( b = \left( {{b}_{1},\ldots ,{b}_{m}}\right) \) , and thus, as far as dealing with derivatives, \( \mathrm{D}f\left( a\right) \) is equal to \( \mathrm{D}{f}_{... | Yes |
Proposition 39.11. Given a normed affine space \( E = \left( {{E}_{1},{a}_{1}}\right) \oplus \cdots \oplus \left( {{E}_{n},{a}_{n}}\right) \), and a normed affine space \( F \), given any open subset \( A \) of \( E \), for any function \( f : A \rightarrow F \), for every \( c \in A \), if \( \mathrm{D}f\left( c\right... | Proof. Since every \( c \in E \) can be written as \( c = a + c - a \), where \( a = \left( {{a}_{1},\ldots ,{a}_{n}}\right) \), defining \( {f}_{a} : \overrightarrow{E} \rightarrow F \) such that, \( {f}_{a}\left( u\right) = f\left( {a + u}\right) \), for every \( u \in \overrightarrow{E} \), clearly, \( \mathrm{D}f\l... | Yes |
For example, consider the function \( f : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2} \), defined such that\n\n\[ f\left( {r,\theta }\right) = \left( {r\cos \left( \theta \right), r\sin \left( \theta \right) }\right) . \] | Then, we have\n\n\[ J\left( f\right) \left( {r,\theta }\right) = \left( \begin{matrix} \cos \left( \theta \right) & - r\sin \left( \theta \right) \\ \sin \left( \theta \right) & r\cos \left( \theta \right) \end{matrix}\right) \]\n\nand the Jacobian (determinant) has value \( \det \left( {J\left( f\right) \left( {r,\the... | Yes |
Consider the quadratic function \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) given by\n\n\[ f\left( x\right) = {x}^{\top }{Ax},\;x \in {\mathbb{R}}^{n}, \]\n\nwhere \( A \) is a real \( n \times n \) symmetric matrix. We claim that\n\n\[ d{f}_{u}\left( h\right) = 2{u}^{\top }{Ah}\;\text{ for all }u, h \in {\mathbb... | Since \( A \) is symmetric, we have\n\n\[ f\left( {u + h}\right) = \left( {{u}^{\top } + {h}^{\top }}\right) A\left( {u + h}\right) \]\n\n\[ = {u}^{\top }{Au} + {u}^{\top }{Ah} + {h}^{\top }{Au} + {h}^{\top }{Ah} \]\n\n\[ = {u}^{\top }{Au} + 2{u}^{\top }{Ah} + {h}^{\top }{Ah}, \]\n\nso we have\n\n\[ f\left( {u + h}\rig... | Yes |
Theorem 39.14. Let \( E, F \), and \( G \), be normed affine spaces, let \( \Omega \) be an open subset of \( E \times F \), let \( f : \Omega \rightarrow G \) be a function defined on \( \Omega \), let \( \left( {a, b}\right) \in \Omega \), let \( c \in G \), and assume that \( f\left( {a, b}\right) = c \) . If the fo... | and if in addition\n\n(6) \( \frac{\partial f}{\partial x} : \Omega \rightarrow \mathcal{L}\left( {\overrightarrow{E};\overrightarrow{G}}\right) \) is also continuous (and thus, in view of (3), \( f \) is \( {C}^{1} \) on \( \Omega \) );\n\nthen\n\n(d) The derivative \( \mathrm{D}g : A \rightarrow \mathcal{L}\left( {\o... | Yes |
Theorem 39.15. Let \( E \) and \( F \) be complete normed affine spaces, let \( A \) be an open subset of \( E \), and let \( f : A \rightarrow F \) be a \( {C}^{1} \) -function on \( A \) . The following properties hold:\n\n(1) For every \( a \in A \), if \( \mathrm{D}f\left( a\right) \) is a linear isomorphism (which... | For every neighborhood \( N \) of a, its image \( f\left( N\right) \) is a neighborhood of \( f\left( a\right) \), and for every open ball \( U \subseteq A \) of center \( a \), its image \( f\left( U\right) \) contains some open ball of center \( f\left( a\right) \) . | No |
Proposition 39.18. Let \( E \) and \( F \) be two normed affine spaces, let \( A \) be an open subset of \( E \), let \( a \in A \), and let \( f : A \rightarrow F \) be a function. If \( \mathrm{D}f\left( a\right) \) exists, then the family of tangent vectors at \( \left( {a, f\left( a\right) }\right) \) to \( \Gamma ... | The proof is actually rather simple. We have \( {T}_{a}\left( \Gamma \right) = a + {T}_{a}\left( \Gamma \right) \), and since \( {T}_{a}\left( \Gamma \right) \) is a subspace of \( \overrightarrow{E} \times \overrightarrow{F} \), the set \( {T}_{a}\left( \Gamma \right) \) is an affine variety. Thus, the affine tangent ... | Yes |
Theorem 39.22. (Taylor-Young) Given two normed affine spaces \( E \) and \( F \) , for any open subset \( A \subseteq E \), for any function \( f : A \rightarrow F \), for any \( a \in A \), if \( {\mathrm{D}}^{k}f \) exists in \( A \) for all \( k \) , \( 1 \leq k \leq m - 1 \), and if \( {\mathrm{D}}^{m}f\left( a\rig... | for any \( h \) such that \( a + h \in A \), and where \( \mathop{\lim }\limits_{{h \rightarrow 0, h \neq 0}}\epsilon \left( h\right) = 0 \) . | Yes |
Theorem 39.23. (Generalized mean value theorem) Let \( E \) and \( F \) be two normed affine spaces, let \( A \) be an open subset of \( E \), and let \( f : A \rightarrow F \) be a function on \( A \) . Given any \( a \in A \) and any \( h \neq 0 \) in \( \overrightarrow{E} \), if the closed segment \( \left\lbrack {a... | \[ \mathop{\max }\limits_{{x \in \left( {a, a + h}\right) }}\begin{Vmatrix}{{\mathrm{D}}^{m + 1}f\left( x\right) }\end{Vmatrix} \leq M \] for some \( M \geq 0 \), then \[ \begin{Vmatrix}{f\left( {a + h}\right) - f\left( a\right) - \left( {\frac{1}{1!}{\mathrm{D}}^{1}f\left( a\right) \left( h\right) + \cdots + \frac{1}{... | Yes |
Proposition 39.26. Given any normed affine space \( E \), for any function \( f : \mathbb{R} \rightarrow \mathbb{R} \) and any function \( g : \mathbb{R} \rightarrow E \), for any \( a \in \mathbb{R} \), letting \( b = f\left( a\right) ,{f}^{\left( i\right) }\left( a\right) = {\mathrm{D}}^{i}f\left( a\right) \), and \(... | \[ {\left( g \circ f\right) }^{\left( m\right) }\left( a\right) = \mathop{\sum }\limits_{{0 \leq j \leq m}}\mathop{\sum }\limits_{\substack{{{i}_{1} + {i}_{2} + \cdots + {i}_{m} = j} \\ {{i}_{1} + {i}_{2},\cdots ,{i}_{m} \geq 0} }}\frac{m!}{{i}_{1}!\cdots {i}_{m}!}{g}^{\left( j\right) }\left( b\right) {\left( \frac{{f}... | Yes |
Proposition 39.28. Given a smooth curve \( \gamma : \rbrack - \eta ,\eta \lbrack \rightarrow E \), letting \( Y \) be the vector field defined on \( \gamma \left( {\rbrack - \eta ,\eta \lbrack }\right) \) such that\n\n\[ Y\left( {\gamma \left( u\right) }\right) = \frac{d\gamma }{dt}\left( u\right) \]\n\nfor any vector ... | The derivative \( {\mathrm{D}}_{Y}X\left( a\right) \) is thus the derivative of the vector field \( X \) along the curve \( \gamma \), and it is called the covariant derivative of \( X \) along \( \gamma \). Given an affine frame \( \left( {O,\left( {{u}_{1},\ldots ,{u}_{n}}\right) }\right) \) for \( \left( {E,\overrig... | Yes |
Proposition 40.1. Let \( E \) be a normed vector space and let \( J : \Omega \rightarrow \mathbb{R} \) be a function, with \( \Omega \) some open subset of \( E \). If the function \( J \) has a local extremum at some point \( u \in \Omega \) and if \( J \) is differentiable at \( u \), then\n\n\[ d{J}_{u} = {J}^{\prim... | Proof. Pick any \( v \in E \). Since \( \Omega \) is open, for \( t \) small enough we have \( u + {tv} \in \Omega \), so there is an open interval \( I \subseteq \mathbb{R} \) such that the function \( \varphi \) given by\n\n\[ \varphi \left( t\right) = J\left( {u + {tv}}\right) \]\n\nfor all \( t \in I \) is well-def... | Yes |
Theorem 40.2. (Necessary condition for a constrained extremum) Let \( \Omega \subseteq {E}_{1} \times {E}_{2} \) be an open subset of a product of normed vector spaces, with \( {E}_{1} \) a Banach space ( \( {E}_{1} \) is complete), let \( \varphi : \Omega \rightarrow {E}_{2} \) be a \( {C}^{1} \) -function (which mean... | Proof. The plan of attack is to use the implicit function theorem; Theorem 39.14. Observe that the assumptions of Theorem 39.14 are indeed met. Therefore, there exist some open subsets \( {U}_{1} \subseteq {E}_{1},{U}_{2} \subseteq {E}_{2} \), and a continuous function \( g : {U}_{1} \rightarrow {U}_{2} \) with \( \lef... | Yes |
Theorem 40.3. (Necessary condition for a constrained extremum in terms of Lagrange multipliers) Let \( \Omega \) be an open subset of \( {\mathbb{R}}^{n} \), consider \( m{C}^{1} \) -functions \( {\varphi }_{i} : \Omega \rightarrow \mathbb{R} \) (with \( 1 \leq m < n) \), let\n\n\[ U = \left\{ {v \in \Omega \mid {\varp... | Proof. The linear independence of the \( m \) linear forms \( d{\varphi }_{i}\left( u\right) \) is equivalent to the fact that the \( m \times n \) matrix \( A = \left( {\left( {\partial {\varphi }_{i}/\partial {x}_{j}}\right) \left( u\right) }\right) \) has rank \( m \) . By reordering the columns, we may assume that ... | Yes |
Proposition 40.4. Let \( E \) be a normed vector space and let \( J : \Omega \rightarrow \mathbb{R} \) be a function, with \( \Omega \) some open subset of \( E \). If the function \( J \) is differentiable in \( \Omega \), if \( J \) has a second derivative \( {\mathrm{D}}^{2}J\left( u\right) \) at some point \( u \in... | Proof. Pick any nonzero vector \( w \in E \). Since \( \Omega \) is open, for \( t \) small enough, \( u + {tw} \in \Omega \) and \( J\left( {u + {tw}}\right) \geq J\left( u\right) \), so there is some open interval \( I \subseteq \mathbb{R} \) such that\n\n\[u + {tw} \in \Omega \;\text{ and }\;J\left( {u + {tw}}\right... | Yes |
Theorem 40.5. Let \( E \) be a normed vector space, let \( J : \Omega \rightarrow \mathbb{R} \) be a function with \( \Omega \) some open subset of \( E \), and assume that \( J \) is differentiable in \( \Omega \) and that \( {dJ}\left( u\right) = 0 \) at some point \( u \in \Omega \) . The following properties hold:\... | Proof. (1) Using the formula of Taylor-Young, for every vector \( w \) small enough, we can write\n\n\[J\left( {u + w}\right) - J\left( u\right) = \frac{1}{2}{\mathrm{D}}^{2}J\left( u\right) \left( {w, w}\right) + \parallel w{\parallel }^{2}\epsilon \left( w\right)\]\n\n\[ \geq \left( {\frac{1}{2}\alpha + \epsilon \lef... | Yes |
Proposition 40.6. For any symmetric matrix \( A \), if \( A \) is positive definite, then there is some \( \alpha > 0 \) such that\n\n\[ \n{x}^{\top }{Ax} \geq \alpha \parallel x{\parallel }^{2}\;\text{ for all }x \in {\mathbb{R}}^{n}.\n\] | Proof. Pick any norm in \( {\mathbb{R}}^{n} \) (recall that all norms on \( {\mathbb{R}}^{n} \) are equivalent). Since the unit sphere \( {S}^{n - 1} = \left\{ {x \in {\mathbb{R}}^{n} \mid \parallel x\parallel = 1}\right\} \) is compact and since the function \( f\left( x\right) = {x}^{\top }{Ax} \) is never zero on \(... | Yes |
Theorem 40.8. (Necessary condition for a local minimum on a convex subset) Let \( J : \Omega \rightarrow \mathbb{R} \) be a function defined on some open subset \( \Omega \) of a normed vector space \( E \) and let \( U \subseteq \Omega \) be a nonempty convex subset. Given any \( u \in U \), if \( {dJ}\left( u\right) ... | Proof. Let \( v = u + w \) be an arbitrary point in \( U \) . Since \( U \) is convex, we have \( u + {tw} \in U \) for all \( t \) such that \( 0 \leq t \leq 1 \) . Since \( {dJ}\left( u\right) \) exists, we can write\n\n\[ \nJ\left( {u + {tw}}\right) - J\left( u\right) = {dJ}\left( u\right) \left( {tw}\right) + \para... | Yes |
Proposition 40.9. (Convexity and first derivative) Let \( f : \Omega \rightarrow \mathbb{R} \) be a function differentiable on some open subset \( \Omega \) of a normed vector space \( E \) and let \( U \subseteq \Omega \) be a nonempty convex subset.\n\n(1) The function \( f \) is convex on \( U \) iff\n\n\[ f\left( v... | Proof. Let \( u, v \in U \) be any two distinct points and pick \( \lambda \in \mathbb{R} \) with \( 0 < \lambda < 1 \) . If the function \( f \) is convex, then\n\n\[ f\left( {\left( {1 - \lambda }\right) u + {\lambda v}}\right) \leq \left( {1 - \lambda }\right) f\left( u\right) + {\lambda f}\left( v\right) \]\n\nwhic... | Yes |
Proposition 40.10. (Convexity and second derivative) Let \( f : \Omega \rightarrow \mathbb{R} \) be a function twice differentiable on some open subset \( \Omega \) of a normed vector space \( E \) and let \( U \subseteq \Omega \) be a nonempty convex subset.\n\n(1) The function \( f \) is convex on \( U \) iff\n\n\[{\... | Proof. First, assume that the inequality in Condition (1) is satisfied. For any two distinct points \( u, v \in U \), the formula of Taylor-Maclaurin yields\n\n\[f\left( v\right) - f\left( u\right) - {df}\left( u\right) \left( {v - u}\right) = \frac{1}{2}{\mathrm{D}}^{2}f\left( w\right) \left( {v - u, v - u}\right)\]\n... | Yes |
On the other hand, if \( f \) is a quadratic function of the form\n\n\[ f\left( u\right) = \frac{1}{2}{u}^{\top }{Au} - {u}^{\top }b \]\n\nwhere \( A \) is a symmetric matrix, we know that\n\n\[ {df}\left( u\right) \left( v\right) = {v}^{\top }\left( {{Au} - b}\right) \] | \[ f\left( v\right) - f\left( u\right) - {df}\left( u\right) \left( {v - u}\right) = \frac{1}{2}{v}^{\top }{Av} - {v}^{\top }b - \frac{1}{2}{u}^{\top }{Au} + {u}^{\top }b - {\left( v - u\right) }^{\top }\left( {{Au} - b}\right) \]\n\n\[ = \frac{1}{2}{v}^{\top }{Av} - \frac{1}{2}{u}^{\top }{Au} - {\left( v - u\right) }^... | Yes |
Theorem 40.11. Given any normed vector space \( E \), let \( U \) be any nonempty convex subset of \( E \). (1) For any convex function \( J : U \rightarrow \mathbb{R} \), for any \( u \in U \), if \( J \) has a local minimum at \( u \) in \( U \), then \( J \) has a (global) minimum at \( u \) in \( U \). | Proof. (1) Let \( v = u + w \) be any arbitrary point in \( U \). Since \( J \) is convex, for all \( t \) with \( 0 \leq t \leq 1 \), we have \[ J\left( {u + {tw}}\right) = J\left( {u + t\left( {v - u}\right) }\right) \leq \left( {1 - t}\right) J\left( u\right) + {tJ}\left( v\right) , \] which yields \[ J\left( {u + {... | Yes |
Theorem 41.1. Let \( X \) be a Banach space, let \( f : \Omega \rightarrow Y \) be differentiable on the open subset \( \Omega \subseteq X \), and assume that there are constants \( r, M,\beta > 0 \) such that if we let\n\n\[ B = \\left\\{ {x \\in X \\mid \\begin{Vmatrix}{x - {x}_{0}}\\end{Vmatrix} \\leq r}\\right\\} \... | A proof of Theorem 41.1 can be found in Ciarlet [41] (Section 7.5). It is not really difficult but quite technical. | No |
Theorem 41.2. Let \( X \) be a Banach space, and let \( f : \Omega \rightarrow Y \) be differentiable on the open subset \( \Omega \subseteq X \) . If \( a \in \Omega \) is a point such that \( f\left( a\right) = 0 \), if \( {f}^{\prime }\left( a\right) \) is a linear isomorphism, and if there is some \( \lambda \) wit... | A proof of Theorem 41.2 can be also found in Ciarlet [41] (Section 7.5). | No |
Proposition 42.1. Given any Euclidean space \( E \) of dimension \( n \), the following properties hold:\n\n(1) Every self-adjoint linear map \( f : E \rightarrow E \) is positive definite iff\n\n\[ \langle f\left( x\right), x\rangle > 0 \]\n\nfor all \( x \in E \) with \( x \neq 0 \) .\n\n(2) Every self-adjoint linear... | Proof. (1) First, assume that \( f \) is positive definite. Recall that every self-adjoint linear map has an orthonormal basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) of eigenvectors, and let \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) be the corresponding eigenvalues. With respect to this basis, for every \( x =... | Yes |
Proposition 42.2. Given a quadratic function\n\n\[ \nQ\left( x\right) = \frac{1}{2}{x}^{\top }{Ax} - {x}^{\top }b \n\]\n\nif \( A \) is symmetric positive definite, then \( Q\left( x\right) \) has a unique global minimum for the solution of the linear system \( {Ax} = b \) . The minimum value of \( Q\left( x\right) \) ... | Proof. Since \( A \) is positive definite, it is invertible, since its eigenvalues are all strictly positive. Let \( x = {A}^{-1}b \), and compute \( Q\left( y\right) - Q\left( x\right) \) for any \( y \in {\mathbb{R}}^{n} \) . Since \( {Ax} = b \), we get\n\n\[ \nQ\left( y\right) - Q\left( x\right) = \frac{1}{2}{y}^{\... | Yes |
Proposition 42.4. If \( A \) is an invertible symmetric matrix, then the function \[ f\left( x\right) = \frac{1}{2}{x}^{\top }{Ax} - {x}^{\top }b \] has a minimum value iff \( A \succcurlyeq 0 \), in which case this optimal value is obtained for a unique value of \( x \), namely \( {x}^{ * } = {A}^{-1}b \), and with \[... | Proof. Observe that \[ \frac{1}{2}{\left( x - {A}^{-1}b\right) }^{\top }A\left( {x - {A}^{-1}b}\right) = \frac{1}{2}{x}^{\top }{Ax} - {x}^{\top }b + \frac{1}{2}{b}^{\top }{A}^{-1}b. \] Thus, \[ f\left( x\right) = \frac{1}{2}{x}^{\top }{Ax} - {x}^{\top }b = \frac{1}{2}{\left( x - {A}^{-1}b\right) }^{\top }A\left( {x - {... | Yes |
Proposition 43.1. If the matrix \( D \) is invertibke, then\n\n\[ \left( \begin{array}{ll} A & B \\ C & D \end{array}\right) = \left( \begin{matrix} I & B{D}^{-1} \\ 0 & I \end{matrix}\right) \left( \begin{matrix} A - B{D}^{-1}C & 0 \\ 0 & D \end{matrix}\right) \left( \begin{matrix} I & 0 \\ {D}^{-1}C & I \end{matrix}\... | The above expression can be checked directly and has the advantage of requiring only the invertibility of \( D \) . | No |
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