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Proposition 15. Let \( \mathrm{X} \) be a topological space on which a topological group \( \mathrm{G} \) operates continuously and transitively. For \( \mathbf{X} \) to be a topological homogeneous space (relative to \( \mathbf{G} \) ) it is sufficient that for some point \( {x}_{0} \in \mathbf{X} \) the mapping \( s ... | Every \( x \in \mathrm{X} \) can be written as \( x = t.{x}_{0} \) for some \( t \in \mathrm{G} \) . If \( \mathrm{V} \) is a neighbourhood of \( e \), then \( \mathrm{V}.x = \left( {\mathrm{V}t}\right) .{x}_{0} \) is a neighbourhood of \( x \) , for we can write \( \left( {\mathrm{V}t}\right) .{x}_{0} = t\left( {\left... | Yes |
Proposition 16. Let \( \mathbf{G} \) be a topological group and let \( \mathbf{H} \) be a normal subgroup of \( \mathbf{G} \). Then the quotient by \( \mathbf{H} \) of the topology of \( \mathbf{G} \) is compatible with the group structure of \( \mathrm{G}/\mathrm{H} \). | If \( x \rightarrow \dot{x} \) is the canonical mapping of \( \mathrm{G} \) onto \( \mathrm{G}/\mathrm{H} \), then we have to show that \( \left( {\dot{x},\dot{y}}\right) \rightarrow \dot{x}{\dot{y}}^{-1} \) is a continuous mapping of \( \left( {\mathrm{G}/\mathrm{H}}\right) \times \left( {\mathrm{G}/\mathrm{H}}\right)... | Yes |
Proposition 17. Let \( \varphi \) be the canonical mapping of a topological group \( \mathbf{G} \) onto a quotient group \( \mathrm{G}/\mathrm{H} \). If \( \mathfrak{V} \) is a fundamental system of neighbourhoods of e in \( \mathrm{G} \), then \( \varphi \left( \mathfrak{V}\right) \) is a fundamental system of neighbo... | This is a particular case of Proposition 5 of Chapter I,§ 5, no. 3. | No |
Proposition 19. If \( \mathrm{H} \) is a discrete normal subgroup of a topological group \( \mathrm{G} \) , then \( \mathrm{G}/\mathrm{H} \) is locally isomorphic to \( \mathrm{G} \) . | Let \( V \) be a neighbourhood of \( e \) in \( G \) which does not contain any point of \( \mathrm{H} \) other than \( e \), and let \( \mathrm{W} \) be a symmetric open neighbourhood of \( e \) in \( \mathrm{G} \) such that \( {\mathrm{W}}^{2} \subset \mathrm{V} \) . Then the restriction to \( \mathrm{W} \) of the ca... | Yes |
Proposition 21. Let \( \mathrm{G} \) be a topological group, \( {\mathrm{G}}_{0} \) a dense subgroup of \( \mathrm{G} \) , \( {\mathrm{H}}_{0} \) a normal subgroup of \( {\mathrm{G}}_{0},\mathrm{H} \) the closure of \( {\mathrm{H}}_{0} \) in \( \mathrm{G} \) and \( ▱ \) the canonical mapping \( \mathrm{G} \rightarrow \... | Since \( {\mathrm{H}}_{0} = \mathrm{H} \cap {\mathrm{G}}_{0} \), it is enough to show that if \( {\mathrm{U}}_{0} \) is any open subset of \( {\mathrm{G}}_{0} \) which is saturated with respect to the relation \( {x}^{-1}y \in {\mathrm{H}}_{0} \), then \( {\mathrm{U}}_{0} \) is the intersection of \( {\mathbf{G}}_{\mat... | Yes |
Proposition 23. A homomorphism \( f \) of a topological group \( \mathbf{G} \) into a topological group \( {\mathrm{G}}^{\prime } \) is continuous on \( \mathrm{G} \) if and only if it is continuous at one point of \( \mathrm{G} \) . | Suppose \( f \) is continuous at a point \( a \in \mathrm{G} \) ; then if \( {\mathrm{V}}^{\prime } \) is any neighbourhood of \( f\left( a\right) ,\mathrm{V} = {\bar{f}}^{1}\left( {\mathrm{\;V}}^{\prime }\right) \) is a neighbourhood of \( a \) . Hence if \( x \) is any point of \( \mathrm{G} \), we have\n\n\[ f\left(... | Yes |
Proposition 24. Let \( f \) be a continuous homomorphism of a topological group \( \mathrm{G} \) into a topological group \( {\mathbf{G}}^{\prime } \) . Then the following three statements are equivalent : a) \( f \) is a strict morphism.\nb) The image under \( f \) of every open set in \( \mathrm{G} \) is an open set ... | In view of Lemma 2 of no. 4, the equivalence of a) and b) follows immediately from the definitions (Chapter I, § 5, no. 3, Proposition 5). The equivalence of b) and c) is a particular case of Proposition 15 of no. 5, if we observe that \( \mathbf{G} \) operates continuously on \( f\left( \dot{\mathbf{G}}\right) \) by t... | Yes |
Proposition 26. The bijection of \( \mathrm{X}/\mathrm{G} \) onto \( \prod \left( {{\mathrm{X}}_{\mathrm{t}}/{\mathrm{G}}_{\mathrm{t}}}\right) \) canonically associated with \( \left( {\varphi }_{t}\right) \) is a homeomorphism. | For since the \( {\varphi }_{t} \) are surjective and open, it follows that \( \varphi = \left( {\varphi }_{t}\right) \) is an open mapping (Chapter I, § 5, no. 3, Corollary to Proposition 8). | Yes |
Proposition 27. Let \( \mathrm{N} \) and \( \mathrm{L} \) be two groups, \( {e}^{\prime } \) and \( {e}^{\prime \prime } \) their respective identity elements. Suppose that we are given a homomorphism \( y \rightarrow {\sigma }_{y} \) of \( \mathbf{L} \) into the group \( \Gamma \) of automorphisms of \( \mathbf{N} \) ... | If \( x,{x}^{\prime },{x}^{\prime \prime } \) are elements of \( \mathrm{N} \) and \( y,{y}^{\prime },{y}^{\prime \prime } \) are elements of \( \mathrm{L} \), then\n\n\[ \left( {\left( {x, y}\right) \left( {{x}^{\prime },{y}^{\prime }}\right) }\right) \left( {{x}^{\prime \prime },{y}^{\prime \prime }}\right) = \left( ... | Yes |
Proposition 28. Let \( \mathrm{L},\mathrm{N} \) be two topological groups, let \( y \rightarrow {\sigma }_{y} \) be a homomorphism of \( \mathbf{L} \) into the group of automorphisms \( \check{\mathbf{\Gamma }} \) of the (non-topological) group structure of \( \mathrm{N} \) ; and suppose that the mapping \( \left( {x, ... | This is an immediate consequence of the definitions and of the properties of the product topology. | No |
Proposition 1. The left and right translations are isomorphisms of the right uniformity onto itself. | As to the right translations, the result is clear, since the relation \( y{x}^{-1} \in \mathrm{V} \) is equivalent to \( \left( {ya}\right) {\left( xa\right) }^{-1} \in \mathrm{V} \) [in other words, the mapping \( \left( {x, y}\right) \rightarrow \left( {{xa},{ya}}\right) \) leaves \( {\mathrm{V}}_{d} \) fixed]. For t... | Yes |
Proposition 2. The symmetry \( x \rightarrow {x}^{-1} \) is an isomorphism of the right uniformity onto the left uniformity. | This is an immediate consequence of Definition 1. | No |
Proposition 3. Every continuous homomorphism \( f \) of a topological group \( \mathrm{G} \) into a topological group \( {\mathbf{G}}^{\prime } \) is uniformly continuous when considered as a mapping of \( {\mathrm{G}}_{d} \) into \( {\mathrm{G}}_{d}^{\prime } \) (or of \( {\mathrm{G}}_{s} \) into \( {\mathrm{G}}_{s}^{... | For if \( {\mathrm{V}}^{\prime } \) is a neighbourhood of the identity element in \( {\mathrm{G}}^{\prime } \), and \( \mathrm{V} = {\bar{f}}^{1}\left( {\mathrm{\;V}}^{\prime }\right) \), then the relation \( y{x}^{-1} \in \mathrm{V} \) implies\n\n\[ f\left( y\right) {\left( f\left( x\right) \right) }^{-1} = f\left( {y... | No |
Proposition 4. If, in a topological group \( \mathbf{G} \), there is a neighbourhood \( \mathbf{V} \) of \( e \) which is complete with respect to either the right or the left uniformity, then \( \mathrm{G} \) is complete. | Suppose for example that \( \mathrm{V} \) is complete with respect to the right uniformity, and let \( \mathfrak{F} \) be a Cauchy filter on \( {\mathrm{G}}_{d} \) ; then \( \mathfrak{F} \) contains a \( {\mathrm{V}}_{d} \) -small set \( \mathrm{M} \), and if \( {x}_{1} \in \mathrm{M} \) we therefore have \( \mathrm{M}... | Yes |
Proposition 5. Let \( {\mathrm{G}}_{1} \) be a topological group, let \( {\mathrm{G}}_{2} \) be a complete Hausdorff topological group, and let \( {\mathrm{H}}_{1} \) (resp. \( {\mathrm{H}}_{2} \) ) be a dense subgroup of \( {\mathrm{G}}_{1} \) (resp. \( {\mathrm{G}}_{2} \) ). Then every continuous homomorphism \( u \)... | \( \bar{u} \) is uniformly continuous with respect to the right uniformities of \( {\mathrm{H}}_{1} \) and \( {\mathrm{H}}_{2} \) (no. I, Proposition 3), hence admits a unique extension to a mapping \( \bar{u} \) of \( {\mathrm{G}}_{1} \) into \( {\mathrm{G}}_{2} \) which is uniformly continuous with respect to the rig... | Yes |
Proposition 6. Let \( \mathfrak{F} \) and \( \mathfrak{G} \) be two Cauchy filters on \( {\mathbf{G}}_{d} \) . Then the image of the filter \( \mathfrak{F} \times \mathfrak{G} \) under the mapping \( \left( {x, y}\right) \rightarrow {xy} \) is a Cauchy filter base on \( {\mathrm{G}}_{d} \) . | Let us evaluate the \ | No |
Proposition 7. Let \( \mathrm{G} \) be a Hausdorff topological group which has a completion \( \widehat{\mathrm{G}} \). Then the closures in \( \widehat{\mathrm{G}} \) of the neighbourhoods of the identity element in \( \mathrm{G} \) form a fundamental system of neighbourhoods of the identity element in \( \widehat{\ma... | Since \( \widehat{\mathrm{G}} \) is regular, every neighbourhood of the identity element in \( \widehat{\mathrm{G}} \) contains the closure \( \mathrm{V} \) of an open neighbourhood \( \mathrm{U} \) of \( e \) in \( \widehat{\mathrm{G}} \), and \( \mathrm{V} \) is also the closure of the trace of \( U \) on \( G \) . | No |
Proposition 8. Let \( \mathbf{G} \) be a topological group which has a Hausdorff completion \( {\widehat{\mathrm{G}}}^{\prime } \) . Then every continuous homomorphism \( u \) of \( \mathrm{G} \) into a complete Hausdorff group \( \mathrm{H} \) can be uniquely factorized into \( u = v \circ \varphi \), where \( v \) is... | Since the kernel of \( u \) is closed and contains \( e \), it contains \( \mathrm{N} \), and hence \( u \) can be written as \( u = w \circ \psi \), where \( w \) is a continuous homomorphism of \( {\mathrm{G}}^{\prime } \) into \( \mathrm{H} \) ; now apply Proposition 5 of no. 3 to \( w \) . | No |
Proposition 9. Let \( \mathrm{G} \) be a commutative group, and let \( {\mathfrak{T}}_{1},{\mathfrak{T}}_{2} \) be two Hausdorff topologies compatible with the group structure of \( \mathrm{G} \) . Suppose that \( {\mathfrak{G}}_{1} \) is finer than \( {\overline{\mathbb{C}}}_{2} \) and that there is a fundamental syst... | Suppose that \( \mathrm{G} \) is written additively. Let \( {\mathcal{U}}_{1} \) be the uniformity on \( \mathrm{G} \) corresponding (no. I) to the topology \( {\mathcal{C}}_{1} \) : it will suffice to show that if \( \mathfrak{F} \) and \( {\mathfrak{F}}^{\prime } \) are two minimal Cauchy filters (Chapter II,§ 3, no.... | Yes |
Corollary 1. Under the hypotheses of Proposition 9, if \( \mathrm{A} \) is a subset of \( \mathrm{G} \) which is a complete subspace with respect to the uniformity \( {\mathcal{U}}_{2} \) corresponding to \( {\mathcal{C}}_{2} \), then \( \mathrm{A} \) is also a complete subspace with respect to the uniformity \( {\math... | If \( {A}_{1} \) is the closure of \( A \) in \( {G}_{1} \), then \( f\left( {A}_{1}\right) \) is contained in the closure of \( \mathrm{A} \) in \( {\mathrm{G}}_{2} \), which by hypothesis is equal to \( \mathrm{A} \) . Since \( f\left( \mathrm{\;A}\right) = \mathrm{A} \) by definition and \( f \) is injective, we hav... | No |
Proposition 1. Let \( \mathbf{G} \) be a topological group operating continuously on a topological space \( \mathrm{X} \), and let \( \mathrm{K} \) be a quasi-compact subset of \( \mathrm{G} \) . Then the mapping \( \rho : \left( {s, x}\right) \rightarrow s.x \) of \( \mathrm{K} \times \mathrm{X} \) into \( \mathrm{X} ... | p \( \xrightarrow[]{\text{ factorizes into }}\mathrm{K} \times \mathrm{X} \rightarrow \mathrm{K} \times \mathrm{X}\overset{{\mathrm{{pr}}}_{s}}{ \rightarrow }\mathrm{X} \), where \( \alpha \left( {s, x}\right) = \left( {s, s.x}\right) .\alpha \) is a homeomorphism, for \( {\alpha }^{-1} : \left( {s, y}\right) \rightarr... | Yes |
Corollary 1. If \( \mathrm{A} \) is a closed (resp. compact) subset of \( \mathrm{X} \), then \( \mathrm{K} \) . \( \mathrm{A} \) is closed in \( \mathrm{X} \) (resp. compact if \( \mathrm{X} \) is Hausdorff). | The assertion concerning closed sets follows from Proposition 1 and the fact that a proper mapping is closed (Chapter I, § 10, no. 1, Proposition 1). The assertion concerning compact sets is trivial. | No |
Corollary 2. If \( \mathrm{K} \) is a quasi-compact subgroup of a topological group \( \mathrm{G} \) , then the equivalence relation \( {x}^{-1}y \in \mathrm{K} \) is closed, and the canonical mapping \( \varphi : \mathrm{G} \rightarrow \mathrm{G}/\mathrm{K} \) is proper. | The first assertion follows from Corollary 1 applied to \( \mathbf{G} \) acting by right translations on itself, and the second assertion then follows from Chapter \( I,§ \) 10, no. 2, Theorem 1. | Yes |
Corollary 3. Let \( \mathbf{K} \) be a quasi-compact normal subgroup of a topological group \( \mathrm{G} \), and let \( \varphi \) be the canonical mapping \( \mathrm{G} \rightarrow \mathrm{G}/\mathrm{K} \). Then for each closed subgroup \( \mathrm{A} \) of \( \mathrm{G} \) the canonical bijection of \( \mathrm{A}/\ma... | Since \( {x}^{-1}y \in \mathrm{K} \) is a closed equivalence relation (Corollary 2), the Corollary follows from Chapter I, § 5, no. 2, Proposition 4. | No |
Proposition 2. Let \( \mathrm{K} \) be a compact group operating continuously on a Hausdorff space X. Then:\n\na) \( \mathrm{K} \) operates properly on \( \mathrm{X} \) . | As to a), since K is compact, \( {\operatorname{pr}}_{2} : \left( {s, x}\right) \rightarrow x \) is proper (Chapter I,§ 10, no. 2, Theorem 1, Corollary 5); hence, \( \mathrm{X} \) being Hausdorff, \( \left( {s, x}\right) \rightarrow \left( {x, s.x}\right) \) is proper (Chapter I, § 10, no. 1, Proposition 5, Corollary 3... | Yes |
Corollary 1. Under the hypotheses of Proposition 2, \( \mathrm{X} \) is compact (resp. locally compact) if and only if \( \mathrm{X}/\mathrm{K} \) is compact (resp. locally compact). | This follows from the fact that the canonical mapping \( \mathrm{X} \rightarrow \mathrm{X}/\mathrm{K} \) is proper, in view of Proposition 9 of Chapter I, \( § \) 10, no. 4. | Yes |
Proposition 3. If a topological group \( \mathbf{G} \) operates properly on a topological space \( \mathrm{X} \), then the orbit space \( \mathrm{X}/\mathrm{G} \) is Hausdorff. If also \( \mathrm{G} \) is Hausdorff, then \( \mathrm{X} \) is Hausdorff. | Let \( \mathrm{C} \subset \mathrm{X} \times \mathrm{X} \) be the graph of the equivalence relation \( \mathrm{R} \) defined by \( \mathrm{G} \) on \( \mathrm{X} \); then \( \mathrm{C} \) is the image of \( \mathrm{G} \times \mathrm{X} \) under the mapping \( \theta : \left( {s, x}\right) \rightarrow \left( {x, s.x}\rig... | Yes |
Proposition 4. Let \( \mathbf{G} \) be a topological group operating properly on a topological space \( \mathrm{X} \), and let \( x \) be a point of \( \mathrm{X} \). Let \( \mathrm{G}.x \) denote the orbit of \( x \), and let \( {\mathbf{K}}_{x} \) denote the stabilizer of \( x \). Then:\n\na) The mapping \( s \righta... | The inverse image of \( \{ x\} \times \mathrm{X} \) under \( \theta : \left( {s, x}\right) \rightarrow \left( {x, s.x}\right) \) is \( \mathrm{G} \times \{ x\} \) ; hence, by Proposition \( 3 \) of Chapter I, \( § \) Io, no. I, the restriction of \( \dot{\theta } \) to \( \mathrm{G} \times \{ x\} \) is a proper mapping... | Yes |
Proposition 5. Let \( \mathrm{G} \) (resp. \( {\mathrm{G}}^{\prime } \) ) be a topological group operating continuously on a topological space \( \mathrm{X} \) (resp. \( {\mathrm{X}}^{\prime } \) ). Let \( ▱ \) be a continuous homomorphism of \( \mathrm{G} \) into \( {\mathrm{G}}^{\prime } \) and let \( \psi \) be a co... | To prove (i), consider the commutative diagram\n\n\[ \mathrm{G} \times \mathrm{X}\overset{\theta }{ \rightarrow }\mathrm{X} \times \mathrm{X} \]\n\n\[ \begin{array}{llllll} \alpha & \downarrow & & & & \downarrow \beta \end{array} \]\n\n\[ {\mathrm{G}}^{\prime } \times {\mathrm{X}}^{\prime }\overset{{\theta }^{\prime }}... | Yes |
Proposition 6. Let \( \mathrm{G} \) be a topological group operating continuously on a topological space \( \mathrm{X} \), and suppose that \( \mathrm{G} \) operates freely on \( \mathrm{X} \). Then \( \mathrm{G} \) operates properly on \( \mathrm{X} \) if and only if the following condition is satisfed:\n\n(FP) The gr... | The set \( \mathrm{C} \) is the image of the mapping \( \theta : \left( {s, x}\right) \rightarrow \left( {x, s.x}\right) \) of \( \mathrm{G} \times \mathrm{X} \) into \( \mathrm{X} \times \mathrm{X} \). We know (Chapter \( \mathrm{I},§ \) 10, no. 1, Proposition 2) that \( \theta \) is proper if and only if \( \mathbf{C... | Yes |
Proposition 7. Let \( \mathrm{G} \) be a locally compact group operating continuously on a Hausdorff space \( \mathrm{X} \) . Then \( \mathrm{G} \) operates properly on \( \mathrm{X} \) if and only if, for each pair of points \( x, y \) of \( \mathrm{X} \), there is a neighbourhood \( {\mathrm{V}}_{x} \) of \( x \) and... | Let \( \mathrm{F} \) be the compact space obtained by adjoining a point at infinity \( \omega \) to \( \mathrm{G} \), and let \( \Gamma \) be the graph of \( \rho : \left( {s, x}\right) \rightarrow s.x \) considered as a subset of \( \mathrm{F} \times \mathrm{X} \times \mathrm{X} \) . Let us show that if the restrictio... | No |
Proposition 8. Let \( \mathrm{G} \) be a discrete group operating properly on a Hausdorff space \( \mathrm{X} \) . Let \( x \) be a point of \( \mathrm{X} \) and let \( {\mathrm{K}}_{x} \) be the stabilizer of \( x \) . Then:\na) The subgroup \( {\mathrm{K}}_{x} \) is finite and there is an open set \( \mathrm{U} \subs... | By Proposition \( 7,{\mathrm{\;K}}_{x} \) is finite. To construct an open set \( \mathrm{U} \) which satisfies the required conditions, notice first that by Proposition 7 there is an open set \( {\dot{\mathrm{U}}}_{0} \) containing \( x \) and such that the set \( \mathrm{K} \) of \( s \in \mathrm{G} \) for which \( s ... | Yes |
Proposition 9. Let \( \mathbf{G} \) be a topological group operating continuously on a locally compact space \( \mathrm{X} \). Then if \( \mathrm{X}/\mathrm{G} \) is Hausdorff it is locally compact. | Since the equivalence relation on \( \mathrm{X} \) defined by \( \mathrm{G} \) is open \( (§2 \), no. 4, Lemma 2), the proposition results from Chapter I, § 10, no. 4, Proposition 10. | No |
Proposition 10. Let \( \mathrm{G} \) be a topological group operating continuously on a locally compact space \( \mathrm{X} \), and suppose that \( \mathrm{X}/\mathrm{G} \) is Hausdorff. Let \( \varphi \) be the canonical mapping of \( \mathrm{X} \) onto \( \mathrm{X}/\mathrm{G} \) . Then if \( {\mathrm{K}}^{\prime } \... | Since the equivalence relation defined by \( \mathbf{G} \) is open (§ 2, no. 4, Lemma 2), the proposition is a particular case of Proposition 10 of Chapter I, \( § \) 10, no. 4. | No |
Proposition 11. Let \( \mathrm{G} \) be a Hausdorff topological group operating properly on a non-empty space \( \mathrm{X} \) . If \( \mathrm{X} \) is compact (resp. locally compact) then so are \( \mathrm{G} \) and \( \mathrm{X}/\mathrm{G} \) . | By hypothesis, the mapping \( \theta : \left( {s, x}\right) \rightarrow \left( {x, s.x}\right) \) of \( \mathrm{G} \times \mathrm{X} \) into \( \mathrm{X} \times \mathrm{X} \) is proper; if \( \mathrm{X} \times \mathrm{X} \) is compact (resp. locally compact) then the Corollary to Proposition 9 of Chapter I,§ 10, no. 4... | No |
Proposition 12. Let \( \mathrm{G} \) be a Hausdorff topological group which operates continuously on a topological space \( \mathrm{X} \). Let \( \ddot{\mathrm{K}} \) be a compact subset of \( \mathrm{X} \), and let \( {\rho }_{\mathbf{K}} \) be the mapping \( \left( {s, x}\right) \rightarrow s.x \) of \( \mathrm{G} \t... | The mapping \( {\widehat{\rho }}_{\mathbf{K}}\;\xrightarrow[]{\text{ factorizes }} \) as \( \;G \times K\overset{{\theta }_{\mathbf{k}}}{ \rightarrow }K \times X\overset{{\rho }_{{\mathbf{r}}_{\mathbf{z}}}}{ \rightarrow }X,\; \) where \( \;{\theta }_{\mathbf{K}} \) is the restriction to \( \mathrm{G} \times \mathrm{K} ... | Yes |
Proposition 13. Let \( \mathrm{G} \) be a locally compact group and let \( \mathrm{H} \) be a closed subgroup of \( \mathrm{G} \). Then the homogeneous space \( \mathrm{G}/\mathrm{H} \) is locally compact and paracompact. | Since \( \mathrm{G}/\mathrm{H} \) is Hausdorff (§ 2, no. 5, Proposition 13) it is locally compact, by Proposition 9 of no. 5 applied to \( \mathrm{H} \) operating on \( \mathrm{G} \) on the right. Thus it remains to show that \( \mathrm{G}/\mathrm{H} \) is paracompact. Let \( \mathrm{V} \) be a symmetric compact neighb... | Yes |
Proposition 14. In a locally compact group \( \mathbf{G} \), the identity component \( \mathbf{C} \) is the intersection of the open subgroups of \( \overline{\mathbf{G}} \) . | C is a closed normal subgroup of G (§ 2, no. 2, Proposition 7), and hence \( \mathrm{G}/\mathrm{C} \) is locally compact (Proposition 13) and totally disconnected (Chapter I, § 11, no. 5, Proposition 9). Since the inverse image of an open subgroup of \( \mathrm{G}/\mathrm{C} \), under the canonical mapping of \( \mathr... | Yes |
Corollary 3. Let \( \mathrm{G} \) be a locally compact group, let \( \mathrm{H} \) be a closed subgroup of \( \mathbf{G} \), and let \( \varphi \) be the canonical mapping of \( \mathbf{G} \) onto \( \mathbf{G}/\mathbf{H} \) . Then the components of \( \mathrm{G}/\mathrm{H} \) are the closures of the images under \( \m... | Let \( \mathrm{C} \) be the identity component of \( \mathrm{G} \) . The components of \( \mathrm{G} \) are the sets \( s\mathrm{C} \), where \( s \in \mathrm{G} \) (§ 2, no. 2, Proposition 7); \( \varphi \left( {s\mathrm{C}}\right) \) is clearly connected, hence so is \( \overline{\varphi \left( {s\mathbf{C}}\right) }... | Yes |
Proposition 2. In a complete group \( \mathrm{G} \), every subfamily of a summable family is summable. | For if Cauchy’s criterion is satisfied by a family \( {\left( {x}_{\iota }\right) }_{\iota \in \mathbf{I}} \), then it is trivially satisfied by every subfamily. | Yes |
Proposition 3. Let \( {\left( {x}_{i}\right) }_{i \in \mathbf{I}} \) be a family of points of a group \( \mathrm{G} \), and let \( {\left( {\mathbf{I}}_{\lambda }\right) }_{\lambda \in \mathbf{L}} \) be a finite partition of \( \mathbf{I} \). If each of the subfamilies \( {\left( {x}_{\imath }\right) }_{\imath \in {\ma... | It is enough to prove the proposition when \( L = \left( {I,2}\right) \); having done so, we then proceed by induction on the number of elements of L. Let \( {s}_{1} = \mathop{\sum }\limits_{{i \in {\mathrm{I}}_{1}}}{x}_{i} \) and \( {s}_{2} = \mathop{\sum }\limits_{{i \in {\mathrm{I}}_{2}}}{x}_{i} \). For each neighbo... | Yes |
Proposition 4. Let \( \mathrm{G} = \mathop{\prod }\limits_{{\lambda \in \mathrm{L}}}{\mathrm{G}}_{\lambda } \) be the product of a family of Hausdorff commutative groups. Then a family \( {\left( {x}_{\iota }\right) }_{\iota \in \mathbf{I}} \) of points of \( \mathbf{G} \) is summable if and only if, for each \( \lambd... | This follows immediately from the condition for convergence, with respect to a filter, of a function which takes its values in a product space (Chapter I, § 1, no. 6, Corollary 1 to Proposition 10); in effect, for each finite subset J of I, we have\n\n\[ \n{\operatorname{pr}}_{\lambda }\left( {\mathop{\sum }\limits_{{\... | Yes |
Proposition 5. Let \( f \) be a continuous homomorphism of a commutative group \( \mathrm{G} \) into a commutative group \( {\mathrm{G}}^{\prime } \). If \( \left( {x}_{\mathrm{t}}\right) \) is a summable family in \( \mathrm{G} \), then \( \left( {f\left( {x}_{t}\right) }\right) \) is a summable family in \( {\mathrm{... | If \( \mathrm{J} \) is any finite subset of the index set, then \( f\left( {\mathop{\sum }\limits_{{\iota \in \mathrm{J}}}{x}_{\iota }}\right) = \mathop{\sum }\limits_{{\iota \in \mathrm{J}}}f\left( {x}_{\iota }\right) \), and the image under \( f \) of a convergent filter base is a convergent filter base (Chapter I, §... | Yes |
Proposition 6. Let \( \left( {x}_{\iota }\right) ,\left( {y}_{\iota }\right) \) be two summable families, with the same index set, in a group G. Then the families \( \left( {-{x}_{i}}\right) ,\left( {n{x}_{i}}\right) \left( {n \in \mathbf{Z}}\right) ,\left( {{x}_{i} + {y}_{i}}\right) \) are summable, and we have\n\n(4)... | For \( x \rightarrow - x \) and \( x \rightarrow {nx} \) are continuous homomorphisms of \( \mathbf{G} \) into \( \mathrm{G} \) ; on the other hand, if \( \left( {x}_{\mathrm{t}}\right) \) and \( \left( {y}_{\mathrm{t}}\right) \) are summable, then the family \( \left( {{x}_{t},{y}_{t}}\right) \) is summable in \( \mat... | Yes |
Proposition 7. If the series defined by the sequences \( \left( {x}_{n}\right) \) and \( \left( {y}_{n}\right) \) are convergent, then so are the series defined by the sequences \( \left( {-{x}_{n}}\right) \) and \( \left( {{x}_{n} + {y}_{n}}\right) \), and we have | \[ \mathop{\sum }\limits_{{n = 0}}^{\infty }\left( {-{x}_{n}}\right) = - \mathop{\int }\limits_{{n = 0}}^{\infty }{x}_{n} \] \[ \mathop{\int }\limits_{{n = 0}}^{\infty }\left( {{x}_{n} + {y}_{n}}\right) = \mathop{\int }\limits_{{n = 0}}^{\infty }{x}_{n} + \mathop{\int }\limits_{{n = 0}}^{\infty }{y}_{n} \] This is an o... | No |
Proposition 8 (Restricted associativity of series). Let \( \left( {k}_{n}\right) \) be a strictly increasing sequence of integers \( \geq 0 \) . If the series whose general term is \( {x}_{n} \) converges, and if we put \( {u}_{n} = \mathop{\sum }\limits_{{p = {k}_{n - 1}}}{x}_{p} \), then the series whose general term... | For the sequence of partial sums of the series \( \left( {u}_{n}\right) \) is a subsequence \( \left( {s}_{{k}_{n} - 1}\right) \) of the sequence \( \left( {s}_{n}\right) \) of partial sums of the series \( \left( {x}_{n}\right) \) . | No |
Proposition 9. The series defined by the sequence \( \left( {x}_{n}\right) \) is commutatively convergent if and only if the sequence \( \left( {x}_{n}\right) \) is summable; and then, for each permutation of \( \mathbf{N} \), we have \[ {\int }_{n = 0}^{\infty }{x}_{\sigma } = \mathop{\sum }\limits_{{n \in \mathbf{N}}... | If the sequence is summable, then clearly the series is commutatively convergent. To prove the reverse implication we shall argue by reductio ad absurdum and suppose that the series \( \left( {x}_{n}\right) \) is commutatively convergent but that the sequence \( \left( {x}_{n}\right) \) is not summable. The image of th... | Yes |
Proposition 1. If \( \mathrm{H} \) is a stable subgroup of a topological group \( \mathrm{G} \) with operators, then the closure \( \overline{\mathrm{H}} \) of \( \mathrm{H} \) in \( \mathrm{G} \) is a stable subgroup of \( \mathrm{G} \) . | We know already (§ 2, no. 1, Proposition 1) that \( \overline{\mathbf{H}} \) is a subgroup of \( \mathbf{G} \). Also if \( \alpha \) is any operator on \( \mathrm{G} \), the image of \( \mathrm{H} \) under the continuous mapping \( x \rightarrow {x}^{\alpha } \) is contained in \( \mathrm{H} \), and therefore the image... | Yes |
Proposition 2. Let \( \mathrm{E} \) be a commutative topological group with operators which is the direct sum of stable subgroups \( {\mathbf{M}}_{i}\left( {\mathrm{I} \leq i \leq n}\right) \) . Let \( {\left( {p}_{i}\right) }_{1 \leq i \leq n} \) be the family of projectors associated with the decomposition \( \mathbf... | Since \( I = \mathop{\sum }\limits_{{i = 1}}^{n}{P}_{i} \) (where \( I \) denotes the identity mapping of \( E \) ) it is sufficient for \( n - 1 \) of the projectors \( {p}_{i} \) to be continuous in order that the \( n \) th should also be continuous. | No |
Proposition 3. Let \( E, F \) be two commutative topological groups with operators, and let \( u \) be a continuous linear mapping of \( \mathbf{E} \) into \( \mathbf{F} \). In order that there should exist a continuous linear mapping \( v \) of \( \mathrm{F} \) into \( \mathrm{E} \) such that \( u \circ v \) is the id... | The conditions are necessary. For we have then \( u\left( {v\left( \mathrm{\;F}\right) }\right) = \mathrm{F} \) and \( a \) fortiori \( u\left( \mathrm{E}\right) = \mathrm{F} \) ; furthermore, if \( p = v \circ u \), then \( p \) is a continuous linear mapping of \( \mathrm{E} \) into itself such that \( {p}^{2} = p \)... | Yes |
Proposition 4. Let \( \;E,\;F\; \) be two commutative topological groups with operators, and let \( u \) be a continuous linear mapping of \( \mathbf{E} \) into \( \mathbf{F} \) . In order that there should exist a continuous linear mapping \( v \) of \( \mathbf{F} \) into \( \mathbf{E} \) such that \( v \circ u \) is ... | The conditions are sufficient, for if they are satisfied we obtain a left inverse \( v \) of \( u \) by taking the composition of the isomorphism of \( u\left( \mathrm{E}\right) \) onto \( \mathrm{E} \) which is the inverse of \( u \), with a continuous projector of \( \mathrm{F} \) onto \( u\left( \mathrm{E}\right) \)... | Yes |
Let \( \mathrm{X} \) be a topological space, and let \( f \) and \( g \) be two mappings of \( \mathrm{X} \) into a topological ring \( \mathrm{A} \). If \( f \) and \( g \) are continuous at a point \( {x}_{0} \in \mathrm{X} \), then \( f + g, - f \) and \( {fg} \) are continuous at this point. | It follows that the continuous mappings of \( \mathrm{X} \) into \( \mathrm{A} \) form a subring of the ring \( {A}^{x} \) of all mappings of \( X \) into \( A \) . We see also that, if \( A \) is commutative, then every polynomial in \( n \) variables, with coefficients in \( \mathbf{A} \) and defined on \( {\mathrm{A... | No |
Proposition 5. Let \( \mathrm{H} \) be a dense subring of a topological ring \( \mathrm{A} \), and let \( \mathrm{K} \) be a subring (resp. left ideal, right ideal, two-sided ideal) of \( \mathrm{H} \) . Then the closure \( \mathrm{K} \) of \( \mathrm{K} \) in \( \mathrm{A} \) is a subring (resp. left ideal, right idea... | The proof is the same as for Proposition 8 of \( §2,\mathrm{n}{.3} \) : if for example \( \mathrm{K} \) is a left ideal in \( \mathrm{H} \), then the mapping \( \left( {z, x}\right) \rightarrow {zx} \) is continuous on \( \widehat{\mathrm{A}} \times \mathrm{A} \) and maps \( \mathrm{H} \times \mathrm{K} \) into \( \mat... | No |
Proposition 6. A Hausdorff topological ring \( \mathrm{A} \) is isomorphic to a dense subring of a complete Hausdorff ring \( \widehat{\mathrm{A}} \), which is determined up to isomorphism (and is called the completion of A). | Let \( \mathrm{A} \) be a topological ring, not necessarily Hausdorff; let \( \mathrm{N} \) be the closure of \( \{ 0\} \) in \( A \), and let \( {A}^{\prime } = A/N \) be the Hausdorff ring associated with \( \mathrm{A} \) . Then \( {\mathrm{A}}^{\prime } \), the completion of \( {\mathrm{A}}^{\prime } \), is called t... | Yes |
Proposition 8. If the additive uniform structure of a topological field \( \mathrm{K} \) is a structure of a complete Hausdorff space, then the multiplicative structure on \( {\mathrm{K}}^{ * } \) is a structure of a complete space. | We shall show that if \( \mathfrak{F} \) is a Cauchy filter with respect to the multiplicative structure on \( {\mathrm{K}}^{ * } \), then \( \mathfrak{F} \) is a Cauchy filter with respect to the additive structure on \( \mathrm{K} \), and does not converge to o; this will establish the result. Let \( \mathrm{U} \) be... | Yes |
Proposition 1. Suppose that the \( {\mathrm{X}}_{\alpha } \) and the \( {\mathrm{G}}_{\alpha } \) satisfy the above hypotheses. a) If the stabilizer of each point of \( {\mathrm{X}}_{\alpha } \) is a compact subgroup of \( {\mathrm{G}}_{\alpha } \) for each \( \alpha \in \mathbf{I} \), then the stabilizer of each point... | Let \( x = \left( {x}_{\alpha }\right) \in \mathrm{X} \), and for each \( \alpha \in \mathrm{I} \) let \( {\mathrm{X}}_{\alpha }^{\prime } = {\mathrm{G}}_{\alpha }.{x}_{\alpha } \) be the orbit of \( {x}_{\alpha } \) . If \( \alpha \leq \beta \), it follows from (I) and the relation \( {g}_{\alpha \beta }\left( {x}_{\b... | Yes |
Corollary 1. If the \( {\mathrm{G}}_{\alpha } \) are compact and the \( {\mathrm{X}}_{\alpha } \) Hausdorff, then the conclusions of \( \mathrm{a} ) and \( \mathrm{b} ) are valid. | For the hypotheses of a) and b) are satisfied, since every closed subgroup of \( {\mathrm{G}}_{\alpha } \) is compact and \( {u}_{\alpha } : {s}_{\alpha } \rightarrow {s}_{\alpha }.{x}_{\alpha } \) is a continuous mapping of a compact space into a Hausdorff space. | No |
Proposition 2. Let \( \mathrm{G} \) be a topological group and let \( {\left( {\mathrm{H}}_{\alpha }\right) }_{\alpha \in \mathrm{I}} \) be a family of normal subgroups of \( \mathrm{G} \) such that \( {\mathrm{H}}_{\alpha } \supset {\mathrm{H}}_{\beta } \) whenever \( \alpha \leq \beta \), and which satisfy the follow... | Clearly the \( {\mathrm{G}}_{\alpha } = \mathrm{G}/{\mathrm{H}}_{\alpha } \) are Hausdorff (§ 2, no. 5, Proposition 13), hence so is \( \widetilde{\mathrm{G}} \) (since it is a subspace of \( \mathop{\prod }\limits_{{\alpha \in I}}{\mathrm{G}}_{\alpha } \) ). The kernel \( \mathrm{H} \) of \( i \) is the intersection o... | Yes |
Corollary 1. If the condition (AP) is satisfied and if in addition the (Hausdorff) groups \( \mathrm{G}/{\mathrm{H}}_{\alpha } \) are complete, then the group \( \mathrm{G} \) has a Hausdorff completion which can be identified with \( \widetilde{\mathrm{G}} \) ; and the mapping \( i : \mathrm{G} \rightarrow \widetilde{... | For \( \widetilde{\mathbf{G}} \) is then complete (no. 2), and Proposition 2 shows that \( i\left( \mathbf{G}\right) \) is isomorphic with the Hausdorff group associated with \( \mathrm{G} \) ; since it is dense in \( \widetilde{G} \), the corollary follows \( \left( {§3,\text{no. 3, Proposition 5}}\right) \) . | Yes |
a) Let \( \mathrm{L} \) be a closed subgroup of \( \mathrm{G} \) ; then, for each \( \alpha \in \mathrm{I} \), the subgroup \( {\mathrm{L}}_{\alpha } = {f}_{\alpha }\left( \mathrm{\;L}\right) \; \) of \( \;{\mathrm{G}}_{\alpha } = \mathrm{G}/{\breve{\mathrm{H}}}_{\alpha } \) is closed, and the isomorphism i of \( \math... | a) Since \( {\mathrm{H}}_{\alpha } \) is compact, \( {\mathrm{{LH}}}_{\alpha } \) is closed in \( \mathrm{G} \) (§ 4, no. 1, Proposition 1, Corollary 1) and therefore \( {\mathrm{L}}_{\alpha } \) is closed in \( {\mathrm{G}}_{\alpha } \) . Since \( i \) identifies the topological groups \( \mathrm{G} \) and \( \mathop{... | Yes |
Proposition 5. Let \( \left( {{\mathrm{G}}_{\alpha },{f}_{\alpha \beta }}\right) \) be an inverse system of Hausdorff topological groups such that the \( {f}_{\alpha \beta } \) are surjective strict morphisms with compact kernels. Then for each \( \alpha \in \mathrm{I} \) the canonical mapping \( {f}_{\alpha } \) of \(... | The facts that \( {f}_{\alpha } \) is surjective and that its kernel is compact are consequences of Chapter I,§ 9, no. 6, Corollary 1 to Proposition 8. It remains to show that \( {f}_{\alpha } \) is a strict morphism. Let \( e \) (resp. \( {e}_{\alpha } \) ) denote the identity element of \( \mathrm{G} \) (resp. \( {\m... | Yes |
Proposition 1. The relation \( y - x \in {\overline{\mathbf{Q}}}_{ + } \) is an ordering on \( \mathbf{R} \) which makes R into a linearly ordered set; is compatible with the additive group structure on \( \mathbf{R} \) , and induces the ordering \( x \leq y \) on \( \overline{\mathbf{Q}} \) . | We begin by showing that the relations \( y - x \in {\overline{\mathbf{Q}}}_{ + } \) and \( z - y \in {\overline{\mathbf{Q}}}_{ + } \) imply \( z - x \in {\overline{\mathbf{Q}}}_{ + } \) . Indeed, the function \( x + y \) is continuous on \( \mathbf{R} \times \mathbf{R} \) , and therefore by (8) we have \( {\overline{\... | Yes |
Proposition 2. Every closed (resp. open) interval in \( \mathbf{R} \) is a closed (resp. open) set in \( \mathbf{R} \) . | The sets \( \left\lbrack {a, \rightarrow \left\lbrack { = a + {\mathbf{R}}_{ + }\text{and}}\right\rbrack \leftarrow, a}\right\rbrack = a - {\mathbf{R}}_{ + } \) are obtained by translation from \( {\mathbf{R}}_{ + } \) and \( - {\mathbf{R}}_{ + } \) respectively and are therefore closed (Chapter III, \( § \) I, no. I);... | Yes |
Proposition 3. As \( r \) runs through the set of rational numbers \( > 0 \), the intervals \( {\mathrm{s}}_{r} = \left\lbrack {-r, + r}\right\rbrack \) in \( \mathbf{R} \) form a fundamental system of neighbourhoods of \( 0 \) . | By Proposition 7 of Chapter III,§ 3, no. 4 we obtain a fundamental system of neighbourhoods of o in \( \mathbf{R} \) by taking the closures in \( \mathbf{R} \) of the intervals\n\n\( {\mathbf{S}}_{r} \cap \mathbf{Q} = \left\lbrack {-r, + r}\right\rbrack \) of \( \mathbf{Q} \) . The proof will be complete if we show tha... | Yes |
Proposition 4. If \( \left( {x, y}\right) \) is any pair of real numbers such that \( x < y \), there is a rational number \( r \) such that \( x < r < y \) . | Since \( \mathbf{Q} \) is dense in \( \mathbf{R} \), it is enough to show that \( \rbrack x, y\lbrack \) is not empty; by translation we may assume \( x = 0 \) and \( y > 0 \) . Now \( \mathbf{R} \) is a Hausdorff space and therefore, by Proposition 3, there is a rational number \( r > 0 \) such that \( y \notin \left\... | No |
Proposition 5. Let \( \mathbf{I} \) be any interval in \( \mathbf{R} \). Then the topology induced on I by the topology of \( \mathbf{R} \) is generated by the open intervals of \( \mathbf{I} \) (where \( \mathbf{I} \) is considered as linearly ordered by the relation \( x \leq y \) ). | Every open interval of \( \mathbf{I} \) is the trace on \( \mathbf{I} \) of an open interval of \( \mathbf{R} \). This is clear for a bounded interval, and the unbounded interval \( \rbrack a, \rightarrow \lbrack \) of \( \mathbf{I} \) is the trace of the unbounded interval \( \rbrack a, \rightarrow \lbrack \) of \( \m... | No |
Every non-empty subset of the real line which is bounded above (resp. bounded below) has a least upper bound (resp. greatest lower bound). | Let \( A \) be a non-empty subset of \( \mathbf{R} \), bounded above, and let \( b \) be an upper bound of \( \mathrm{A} \), so that \( \mathrm{A} \subset \mathrm{j} \leftarrow, b\rbrack \) . For each \( x \in \mathrm{A} \) consider the set \( {\mathrm{A}}_{x} \) of numbers \( \geq x \) which belong to \( \mathrm{A} \)... | Yes |
Proposition 1. A non-empty subset \( \mathbf{A} \) of \( \mathbf{R} \) is an interval if and only if, whenever \( a \) and \( b \) are any two points of \( \mathrm{A} \) such that \( a < b \), the closed interval \( \left\lbrack {a, b}\right\rbrack \) is contained in \( \mathrm{A} \) . | The condition is clearly necessary. Conversely, suppose that it is satisfied. If \( A \) is neither bounded above nor below it must be the whole of \( \mathbf{R} \) , for if \( x \) is any point of \( \mathbf{R} \) there are then two points \( a, b \) of \( \mathrm{A} \) such that \( a < x < b \) . If \( \mathrm{A} \) ... | Yes |
Proposition 2. Every non-empty open set in \( \mathbf{R} \) is the union of a countable family of mutually disjoint open intervals. | Let \( \mathbf{A} \) be a non-empty open set in \( \mathbf{R} \) . Since \( \mathbf{R} \) is locally connected, every component of \( \mathrm{A} \) is a connected open set (Chapter I,§ 11, no. 6, Proposition 11) and therefore an open interval by Theorem 4. Any two of these open intervals are disjoint; on the other hand... | Yes |
Proposition 1. The functions \( {xy} \) and \( \mathbf{I}/x \), defined respectively on \( \mathbf{Q} \times \mathbf{Q} \) and \( {\mathbf{Q}}^{ * } \), can be extended by continuity to \( \mathbf{R} \times \mathbf{R} \) and \( {\mathbf{R}}^{ * } \) respectively, and define a field structure on \( \mathbf{R} \) . Endow... | All the properties of topological fields established in \( §6 \) of Chapter III are of course applicable; in particular, every rational function of \( n \) real variables, with real coefficients, is continuous at every point of \( {\mathbf{R}}^{n} \) where its denominator does not vanish. | No |
Proposition 2. The multiplicative group \( {\mathbf{R}}^{ * } \) of real numbers \( \neq 0 \) is a topological group isomorphic to the product of its subgroups \( {\mathbf{R}}_{ + }^{ * } \) and \( {\mathrm{U}}_{0} \), where\n\n\[ \n{U}_{0} = \{ - 1, + 1\}\n\] | For each \( x \neq 0 \) let \( \operatorname{sgn}x \) denote \( \frac{x}{\left| x\right| } \) (sign of \( x \) ). The function sgn is a homomorphism of \( {\mathbf{R}}^{ * } \) onto \( {U}_{0} \) . We have \( x = \left| x\right| \operatorname{sgn}x \), and this decomposition of \( x \) as the product of an element of \... | Yes |
Proposition 1. All non-empty open intervals of \( \mathbf{R} \) are homeomorphic to \( \mathbf{R} \) . | Consider first a bounded open interval \( \mathrm{I} = \rbrack a, b\left\lbrack \right. \left( {a < b}\right) \) . For each \( x \in \mathrm{I} \) put \( f\left( x\right) = - \left( {\frac{1}{x - a} + \frac{1}{x - b}}\right) \) . This function is continuous and strictly increasing on \( \mathrm{I} \), for we have seen ... | Yes |
Proposition 2. Every homeomorphism \( f \) of \( \mathbf{R} \) onto an interval \( \rbrack a, b\lbrack \) can be extended to a homeomorphism \( \bar{f} \) of \( \overline{\mathbf{R}} \) onto \( \left\lbrack {a, b}\right\rbrack \) . If \( f \) is an increasing function, then \( \bar{f} \) is an order isomorphism of \( \... | Let \( f \) be an increasing homeomorphism. If we extend \( f \) to \( \mathbf{R} \) by putting \( \bar{f}\left( {-\infty }\right) = a \) and \( \bar{f}\left( {+\infty }\right) = b \), it is obvious that \( \bar{f} \) is a strictly increasing mapping (and therefore a bijection) of \( \overline{\mathbf{R}} \) onto \( \l... | Yes |
Proposition 3. The extended real line is compact. | Hence (Chapter II,§ 4, no. 1, Theorem 1) there is a unique uniformity on \( \overline{\mathbf{R}} \) compatible with its topology; this uniformity is isomorphic with the uniformity induced on \( \left\lbrack {a, b}\right\rbrack \) by the additive uniformity of \( \mathbf{R} \) . But it should be remarked that the unifo... | No |
Proposition 7. The function \( x + y \) can be extended by continuity to each of the sets \( {\mathrm{A}}^{\prime } \times {\mathrm{A}}^{\prime } \) and \( {\mathrm{A}}^{\prime \prime } \times {\mathrm{A}}^{\prime \prime } \), according to the formulae\n\n(2)\n\n\[ \left\{ \begin{array}{ll} x + \left( {+\infty }\right)... | Let us show, for example, that as \( \left( {x, y}\right) \) tends to the point \( \left( {a, + \infty }\right) \) \( \left( {a \neq - \infty }\right) \) while remaining in \( \mathbf{R} \times \mathbf{R}, x + y \) tends to \( + \infty \) . There exists a finite number \( b < a \), and the interval \( \rbrack b, + \inf... | Yes |
Proposition 2. Let \( f \) and \( g \) be two real-valued functions defined on a set \( \mathbf{X} \) filtered by a filter \( \mathfrak{F} \). If \( {\lim }_{\mathfrak{F}}f \) and \( {\lim }_{\mathfrak{F}}g \) exist, and if \( {\lim }_{\mathfrak{F}}f > {\lim }_{\mathfrak{F}}g \), then there is a set \( \mathrm{A} \in \... | Let \( a = \mathop{\lim }\limits_{\mathfrak{F}}f \), let \( b = \mathop{\lim }\limits_{\mathfrak{F}}g \) and let \( c \) be such that \( b < c < a \). The interval \( \rbrack c, + \infty \rbrack \) of \( \overline{\mathbf{R}} \) (resp. \( \lbrack - \infty, c\lbrack \) ) is a neighbourhood of \( a \) (resp. \( b \) ); h... | Yes |
Proposition 5. Let \( f \) be a real-valued function defined on a set \( \mathrm{X} \). On the set \( \mathfrak{F}\left( \mathrm{X}\right) \) of all finite subsets of \( \mathrm{X} \), directed with respect to the relation \( \mathrm{c} \), the real-valued function \( \mathrm{H} \rightarrow \mathop{\sup }\limits_{{x \i... | Let \( \varphi \left( \mathrm{H}\right) = \mathop{\sup }\limits_{{x \in \mathrm{H}}}f\left( x\right) \). Clearly \( \varphi \) is increasing, and therefore has a limit \( a \) (no. 2, Theorem 2); and since \( \varphi \left( \mathrm{H}\right) \leq \mathop{\sup }\limits_{{x \in A}}f\left( x\right) \) for all \( \mathrm{H... | Yes |
Proposition 10. In the product space \( \overline{\mathbf{R}}\mathbf{x} \) the upper envelope \( \sup {f}_{t} \) of a family of real-valued functions \( {\left( {f}_{i}\right) }_{i \in \mathbf{I}} \) is the limit, with respect to the directed set \( \mathfrak{F}\left( \mathbf{I}\right) \) of finite subsets of \( \mathb... | This follows immediately from Proposition 5 of no. 4 and from Chapter I, \( §7 \), no. 6, Corollary 1 to Proposition 10. | No |
Corollary 2. If \( \mathfrak{H} \) is a filter finer than \( \mathfrak{G} \), we have\n\n\[ \mathop{\operatorname{lim\ inf}}\limits_{\varnothing }f \leq \mathop{\operatorname{lim\ inf}}\limits_{\varnothing }f \leq \mathop{\operatorname{lim\ sup}}\limits_{\varnothing }f \leq \mathop{\operatorname{lim\ sup}}\limits_{\var... | For every cluster point of \( f \) with respect to \( \mathfrak{H} \) is also a cluster point of \( f \) with respect to \( \mathcal{G} \) (Chapter I,§ 7, no. 3). In particular, if \( \mathop{\lim }\limits_{\mathfrak{H}}f \) exists, then\n\n\[ \mathop{\lim }\limits_{f}\overline{\operatorname{f}} \leq \mathop{\lim }\lim... | No |
Proposition 11. Let \( f \) and \( g \) be two real-valued functions defined on a filtered set \( \mathrm{X} \). Then the relation \( f \leq g \) implies\n\n\[ \left\{ \begin{array}{l} \lim \sup f \leq \lim \sup g, \\ \lim \inf f \leq \lim \inf g. \end{array}\right. \] | This is an immediate consequence of the relations (I2). | No |
Proposition 12. Let \( f \) and \( g \) be two real-valued functions defined on a set \( \mathbf{X} \) , and let \( \mathrm{A} \) be a non-empty subset of \( \mathrm{X} \) .\n\n(i) We have\n\n(17)\n\n\[ \mathop{\sup }\limits_{{x \in \mathbf{A}}}\left( {f\left( x\right) + g\left( x\right) }\right) \leq \mathop{\sup }\li... | Let \( \mathbf{H} \) be any finite subset of \( \mathbf{A} \) . If \( {x}_{0} \) is one of the points of \( \mathbf{H} \) where \( f + g \) takes its greatest value, then we have\n\n\[ f\left( {x}_{0}\right) + g\left( {x}_{0}\right) \leq \mathop{\sup }\limits_{{x \in \mathbf{H}}}f\left( x\right) + \mathop{\sup }\limits... | Yes |
Proposition 1. A real-valued function \( f \) on a topological space \( \mathrm{X} \) is lower semi-continuous if and only if, for each finite real number \( k,{\overrightarrow{f}}^{t}\left( {\rbrack k, + \infty }\right) \) [the set of all \( x \in \mathrm{X} \) such that \( f\left( x\right) > k \) ] is an open set in ... | For this condition shows that \( \overrightarrow{f}\left( {\rbrack k, + \infty \rbrack }\right) \) is a neighbourhood of each of its points. | No |
Proposition 2. Let \( f \) and \( g \) be two real-valued functions, lower semi-continuous at a point \( a \in \mathrm{X} \). Then the functions \( \inf \left( {f, g}\right) \) and \( \sup \left( {f, g}\right) \) are lower semi-continuous at a; so is \( f + g \) whenever it is defined, and so is \( {fg} \) if \( f \) a... | We give the proof for \( f + g \) ; the argument is analogous in the other cases. The result is clear if either \( f\left( a\right) \) or \( g\left( a\right) \) is equal to \( - \infty \) ; if not, then \( f\left( a\right) + g\left( a\right) > - \infty \). Every finite number \( h < f\left( a\right) + g\left( a\right) ... | Yes |
Proposition 3. A real-valued function \( f \), defined on a topological space \( \mathrm{X} \) , is lower semi-continuous at a point \( a \in \mathrm{X} \) if and only if \( \mathop{\liminf }\limits_{{x \rightarrow a}}f\left( x\right) = f\left( a\right) \n\n[or, equivalently, if and only if \( \mathop{\liminf }\limits_... | The condition is necessary. For, given any \( h < f\left( a\right) \), there is a neighbourhood \( \mathrm{V} \) of \( a \) such that \( h < f\left( x\right) \) for all \( x \in \mathrm{V} \) ; therefore\n\n\[ \nh \leq \mathop{\inf }\limits_{{x \in \mathrm{v}}}f\left( x\right) \leq \mathop{\liminf }\limits_{{x \rightar... | Yes |
Proposition 4. Let \( f \) be any real-valued function defined on a dense subset \( \mathbf{A} \) of a topological space \( \mathrm{X} \) . If, for each \( x \in \mathrm{X} \), we put \( g\left( x\right) = \liminf f\left( y\right) \) , then \( g \) is lower semi-continuous on \( \mathrm{X} \) . | For, given any \( h < g\left( x\right) \), there is an open neighbourhood \( \mathrm{V} \) of \( x \) such that, for all \( z \in \mathrm{V} \cap \mathrm{A} \), we have \( h < f\left( z\right) \) ; now \( \mathrm{V} \) is a neighbourhood of each of its points \( y \) ; thus we have \( \liminf f\left( z\right) = g\left(... | Yes |
Proposition 1. If the families \( {\left( {x}_{\lambda }\right) }_{\lambda \in \mathrm{L}} \) and \( {\left( {y}_{\mu }\right) }_{\mu \in \mathrm{M}} \) of finite real numbers are summable in \( \mathbf{R} \), then so is the family \( {\left( {x}_{\lambda }{y}_{\mu }\right) }_{\left( {\lambda ,\mu }\right) \in \mathbf{... | Every finite subset of \( \mathbf{L} \times \mathbf{M} \) is contained in a finite subset of the form \( \mathrm{H} \times \mathrm{K} \), where \( \mathrm{H} \) is a finite subset of \( \mathrm{L} \) and \( \mathrm{K} \) is a finite subset of \( \mathrm{M} \) . By hypothesis, there exists a number \( a > 0 \) such that... | Yes |
Proposition 2. Every family \( \left( {x}_{\iota }\right) \) of real numbers \( \geq 0 \) is summable in \( \overline{\mathbf{R}} \) . | For the mapping \( \mathrm{H} \rightarrow {s}_{\mathrm{H}} \) of the directed set \( \mathfrak{F}\left( \mathrm{I}\right) \) into \( \overline{\mathbf{R}} \) is increasing; hence \( \left( {§5\text{, no. 2, Theorem 2}}\right) \) has a limit. | Yes |
Proposition 3. Every family \( \left( {1 + {u}_{\iota }}\right) \) [resp. \( \left( {1 - {u}_{\iota }}\right) \) ] of numbers \( \geq 1 \) (resp. \( \geq \) o and \( \leq \) I) is multipliable in \( \overline{\mathbf{R}} \) . | Same proof as for Proposition 2. | No |
Proposition 4. A series of finite real numbers is commutatively convergent if and only if it is absolutely convergent. | This follows from Chapter III, \( §5 \), no. 7, Proposition 9 and from Theorem 3 of no. 2. | No |
Proposition 5. An infinite product of finite real numbers is commutatively convergent if and only if it is absolutely convergent. | This follows from Chapter III,§ 5, no. 7, Proposition 9, and from Theorem 4 above. | No |
If \( \mathbf{u} \sim \left( {1,2,3}\right) \) and \( \mathbf{v} \sim \left( {-3,1, - 2}\right) \), compute \( \mathbf{u} \cdot \mathbf{v} \) and the enclosed angle. | From (1.13)\n\n\[ \mathbf{u} \cdot \mathbf{v} = \left( 1\right) \left( {-3}\right) + \left( 2\right) \left( 1\right) + \left( 3\right) \left( {-2}\right) = - 7 \]\n\nand from (1.8)\n\n\[ \theta = {\cos }^{-1}\frac{\mathbf{u} \cdot \mathbf{v}}{\left| \mathbf{u}\right| \left| \mathbf{v}\right| },\;0 \leq \theta \leq \pi ... | Yes |
Determine a unit vector \( \mathbf{e} \) mutually \( \bot \) to the vectors \( \mathbf{a} \sim \left( {1, - 2,3}\right) \) and \( \mathbf{b} \sim \left( {-1,0,1}\right) \) using and without using the cross product. | In terms of the cross product, the vector we seek is\n\n\[ \mathbf{e} = \pm \frac{\mathbf{a} \times \mathbf{b}}{\left| \mathbf{a} \times \mathbf{b}\right| } \]\n\nFrom (1.25),\n\n\[ \mathbf{a} \times \mathbf{b} = \left| \begin{matrix} {\mathbf{e}}_{x} & {\mathbf{e}}_{y} & {\mathbf{e}}_{z} \\ 1 & - 2 & 3 \\ - 1 & 0 & 1 ... | Yes |
If \( \mathbf{v} \sim \left( {{v}_{x},{v}_{y},{v}_{z}}\right) \) and \( \mathbf{{Tv}} \sim \left( {-2{v}_{x} + 3{v}_{z}, - {v}_{z},{v}_{x} + 2{v}_{y}}\right) \), determine the Cartesian components of \( {\mathbf{T}}^{T}\mathbf{v} \) . | Let \( {\mathbf{T}}^{\mathbf{T}}\mathbf{v} \sim \left( {a, b, c}\right) \) and \( \mathbf{u} \sim \left( {\alpha ,\beta ,\gamma }\right) \) . By definition, \( \mathbf{u} \cdot {\mathbf{T}}^{\mathbf{T}}\mathbf{v} = \mathbf{v} \cdot \mathbf{T}\mathbf{u} \) for all vectors \( \mathbf{u} \) and \( \mathbf{v} \) . That is,... | Yes |
Determine the Cartesian components of the tensor \( \mathbf{T} \) defined in Problem 1.4. | Applying \( T \) successively to \( {\mathbf{e}}_{x},{\mathbf{e}}_{y} \), and \( {\mathbf{e}}_{z} \), we have\n\n\[ \n{\mathbf{{Te}}}_{x} \sim \left( {-2,0,1}\right) = - 2\left( {1,0,0}\right) \; + \left( {0,0,1}\right) \n\]\n\n\[ \n{\mathbf{{Te}}}_{y} \sim \left( {0,0,2}\right) \; = \; + 2\left( {0,0,1}\right) \n\]\n\... | Yes |
If \( \mathbf{u} \sim \left( {{u}_{x},{u}_{y},{u}_{z}}\right) \), then \( \mathbf{u} \times \) may be regarded as a 2nd order tensor whose action on any vector \( \mathbf{v} \sim \left( {{v}_{x},{v}_{y},{v}_{z}}\right) \) is defined by (1.24). Find the Cartesian components of \( \mathbf{u} \times \) . | Applying \( \mathbf{u} \times \), successively, to \( {\mathbf{e}}_{x},{\mathbf{e}}_{y} \), and \( {\mathbf{e}}_{z} \), we have, according to (1.24),\n\n\[ \mathbf{u} \times {\mathbf{e}}_{x} \sim \left( {0,{u}_{z}, - {u}_{y}}\right) = \;{u}_{z}\left( {0,1,0}\right) - {u}_{y}\left( {0,0,1}\right) \]\n\n\[ \mathbf{u} \ti... | Yes |
Given\n\n\[ \n{\mathbf{g}}_{1} \sim \left( {1, - 1,2}\right) ,{\mathbf{g}}_{2} \sim \left( {0,1,1}\right) ,{\mathbf{g}}_{3} \sim \left( {-1, - 2,1}\right) \]\n\n\[ \n\mathbf{v} \sim \left( {3,3,6}\right) \]\n\nfind \( \left( {{v}^{1},{v}^{2},{v}^{3}}\right) \) . | ## Solution.\n\nEquating corresponding Cartesian components on both sides of (2.1), i.e. applying rules (1.5)-(1.7), we get\n\n\[ \n3 = {v}^{1}\; - {v}^{3} \]\n\n\[ \n3 = - {v}^{1} + {v}^{2} - 2{v}^{3} \]\n\n\[ \n6 = 2{v}^{1} + {v}^{2} + {v}^{3} \]\n\nSolving these simultaneous linear algebraic equations, we have\n\n\[... | Yes |
Compute the cellar components of the vector \( \mathbf{v} \) given in Problem 2.1. | Writing (2.5) in extended form and noting (2.4), we have\n\n\[ \mathbf{v} \cdot {\mathbf{g}}_{1} = \left( {{v}_{1}{\mathbf{g}}^{1} + {v}_{2}{\mathbf{g}}^{2} + {v}_{3}{\mathbf{g}}^{3}}\right) \cdot {\mathbf{g}}_{1} \]\n\n\[ = {v}_{1}{\mathbf{g}}^{1} \cdot {\mathbf{g}}_{1} + {v}_{2}{\mathbf{g}}^{2} \cdot {\mathbf{g}}_{1}... | Yes |
Given \( \mathbf{u} = 2{\mathbf{g}}_{1} - {\mathbf{g}}_{2} + 4{\mathbf{g}}_{3} \) and \( \mathbf{w} = - 3{\mathbf{g}}^{1} + 2{\mathbf{g}}^{2} - 2{\mathbf{g}}^{3} \), compute \( \mathbf{u} \cdot \mathbf{v},\mathbf{w} \cdot \mathbf{v},\mathbf{w} \cdot \mathbf{v} \), and \( \mathbf{u} \cdot \mathbf{w} \), where \( \mathbf... | The roof components of \( \mathbf{u} \) are given, while the cellar components of \( \mathbf{v} \) are given in the solution in Problem 2.3. Thus, by (2.9),\n\n\[ \mathbf{u} \cdot \mathbf{v} = \left( 2\right) \left( {12}\right) + \left( {-1}\right) \left( 9\right) + \left( 4\right) \left( {-3}\right) = 3. \]\n\nThe cel... | Yes |
Given \( \mathbf{u} = 2{\mathbf{g}}_{1} - {\mathbf{g}}_{2} + 4{\mathbf{g}}_{3},\mathbf{v} = 2{\mathbf{g}}_{1} + 3{\mathbf{g}}_{2} - {\mathbf{g}}_{3} \), where the basis \( \left\{ {\mathbf{g}}_{i}\right\} \) is given in Problem 2.1, compute the cellar components of \( \mathbf{u} \times \mathbf{v} \) . | Let us write (2.12) in extended form for \( k = 1 \), taking note of (2.14).\n\n\[ \n{\left( \mathbf{u} \times \mathbf{v}\right) }_{1} = {\epsilon }_{ij1}{u}^{i}{v}^{j} = {\epsilon }_{111}{u}^{1}{v}^{1} + {\epsilon }_{121}{u}^{1}{v}^{2} + {\epsilon }_{131}{u}^{1}{v}^{3} \n\] \n\n\[ \n+ {\mathcal{G}}_{211}{u}^{2}{v}^{1}... | Yes |
Compute the cellar, roof, and mixed components of the tensor given in Problem 1.4, using the base vectors given in Problem 2.1 and the associated reciprocal base vectors given in the Solution to Problem 2.2. | We are given that\n\n\[ \mathrm{{Tv}} \sim \left( {-2{v}_{x} + 3{v}_{z}, - {v}_{z},{v}_{x} + 2{v}_{y}}\right) \]\n\nand\n\n\[ {\mathbf{g}}_{1} \sim \left( {1, - 1,2}\right) ,{\mathbf{g}}_{2} \sim \left( {0,1,1}\right) ,{\mathbf{g}}_{3} \sim \left( {-1, - 2,1}\right) .\n\nHence,\n\n\[ {\mathbf{{Tg}}}_{1} \sim \left( {4,... | Yes |
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