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Theorem 2 (Formal criteria for the existence of an adjoint). A functor \( G : A \rightarrow X \) has a left adjoint if and only if the right Kan extension \( {\operatorname{Ran}}_{G}{1}_{A} : X \rightarrow A \) exists and is preserved by \( G \) ; when this is the case, this right Kan extension is a left adjoint \( F =... | Proof. If \( G \) has a left adjoint \( F \), with unit \( \eta : {1}_{X} \rightarrow {GF} \) and counit \( \varepsilon : {FG} \rightarrow {1}_{A} \), then we can construct for all functors \( H : A \rightarrow C \) (in particular, for the identity functor \( {1}_{A} \) ) a bijection\n\n\[ \operatorname{Nat}\left( {S,{... | Yes |
Proposition 3. If \( G : A \rightarrow X \) has a left adjoint \( F \) with counit \( \varepsilon : {FG} \rightarrow 1 \) , then \( {\operatorname{Ran}}_{G}{1}_{A} \) exists, is equal to \( F \) with counit \( \varepsilon \), and is preserved by any functor whatever. | Now suppose conversely that \( {1}_{A} \) has a right Kan extension \( R \) along \( G \) , and that this extension is preserved by \( G \) . We then have bijections\n\n\[ \varphi = {\varphi }_{S} : \operatorname{Nat}\left( {S, R}\right) \; \cong \operatorname{Nat}\left( {{SG},{1}_{A}}\right) ,\;\varphi \left( {S\overs... | Yes |
Theorem 1. In each symmetric monoidal category \( M \) there is a function which assigns to each pair \( \left( {{v\sigma },{w\tau }}\right) \) of permuted words of the same length \( n \) a (unique) natural isomorphism\n\n\[ \n{\operatorname{can}}_{M}\left( {{v\sigma },{w\tau }}\right) : {\left( v\sigma \right) }_{M} ... | Proof. There is always at least one such map between different permuted words, since we can use instances of \( \alpha \) to rearrange the parentheses and instances of \( \gamma \) to transpose adjacent arguments. This will provide for any desired permutations of the arguments, since all the permutations of the symmetr... | No |
Theorem 1. Any monoidal category \( M \) is categorically equivalent, via a strong monoidal functor \( G : M \rightarrow S \) and a strong monoidal functor \( F : S \rightarrow M \), to a strict monoidal category \( S \) . | Proof. The coherence theorem yields unique \ | No |
Theorem 2. If the monoidal category \( M \) is equivalent by a strong monoidal functor \( G : M \rightarrow S \) to a strict monoidal category \( S \), then coherence holds for the associativity of the tensor product \( ▱ \) in \( M \) . | Proof. Suppose that \( v \) and \( w \) are two tensor words in \( k \) letters, while \( \theta \) and \( {\theta }^{\prime } : v \rightarrow w \) are two natural transformations between the corresponding functors, both constructed as combinations of the associativity transformation \( \alpha \) in \( M \) . Now use t... | Yes |
Proposition 2.1. The mapping \( \rho \) of \( \mathcal{E} + \mathcal{F} \) into \( \mathcal{E} \times \mathcal{F} \) is bilinear, and the range of \( \rho \) generates \( \mathcal{E} \times \mathcal{F} \). Moreover, if \( \varphi \) is any bilinear transformation of \( \mathcal{E} + \mathcal{F} \) into a linear space \... | Proof. It is clear from (4) and the definition that \( \rho \) is a bilinear transformation, and it has just been observed that every element of \( \mathcal{E}\dot{ \times }\mathcal{F} \) can be written (not uniquely) as a finite sum of decomposable elements. Thus the range of \( \rho \) generates \( \mathcal{E}\dot{ \... | Yes |
Proposition 2.2. Let \( \mathcal{E} \) and \( \mathcal{F} \) be complex [real] linear spaces and suppose given a pair \( \left( {\mathcal{G},\sigma }\right) \), where \( \mathcal{G} \) is a complex [real] linear space and \( \sigma \) is a bilinear transformation of \( \mathcal{E} + \mathcal{F} \) into \( \mathcal{G} \... | Proof. We simply define \( \Phi \) to be the result of factoring \( \rho \) through \( \mathcal{G} \) as in (ii), and denote by \( \Psi \) the result of factoring \( \sigma \) through \( \mathcal{E} \times \mathcal{F} \). It is clear that \( \Phi \) and \( \Psi \) are mutually inverse mappings on the ranges of \( \sigm... | Yes |
Proposition 2.3. Let \( \mathcal{E} \) and \( \mathcal{F} \) be linear spaces over the same scalar field, let \( X \) be a subset of \( \mathcal{E} \), let \( Y \) be a subset of \( \mathcal{F} \), and let \( T = \{ x \otimes y : x \in X, y \in Y\} \) . If \( X \) and \( Y \) generate \( \mathcal{E} \) and \( \mathcal{... | Proof. The first assertion is obvious, and is included here only for completeness, while the third assertion is an immediate consequence of the first two. Thus the proof comes down to showing that \( T \) is linearly independent when \( X \) and \( Y \) are, and here it is clearly enough to treat the case in which \( X... | Yes |
Proposition 3.6. If \( C \) is connected subset of a topological space \( X \), and if \( A \) is a subset of \( X \) such that \( C \cap \partial A = \varnothing \), then either \( C \subset A \) or \( C \cap A = \varnothing \) . | Proof. If \( C \cap \partial A = \varnothing \) and \( x \in C \cap A \), then \( x \in {A}^{ \circ } \) . Thus \( C \cap A = C \cap {A}^{ \circ } \) is open relative to \( C \) . Since \( A \) and \( X \smallsetminus A \) have the same boundary, this shows that \( C \smallsetminus A \) is also open relative to \( C \)... | Yes |
Proposition 3.7. If \( E \) is an arbitrary subset of the real line \( \mathbb{R} \), then the following conditions are equivalent:\n\n(i) \( E \) is connected,\n\n(ii) If \( a, b \in E \), and if \( a < c < b \), then \( c \in E \),\n\n(iii) \( E \) is either an interval (open, closed, or half-open), or a ray (open or... | Proof. That (iii) implies (i) was proved in Example I, while the proof that (ii) implies (iii) amounts to nothing more than a careful consideration of cases. Finally, to show that (i) implies (ii) we note that \( \left( {-\infty, c}\right) \cup \left( {c, + \infty }\right) \) is a disconnection of the set \( \mathbb{R}... | No |
Proposition 3.8. If \( U \) is a connected open subset of \( {\mathbb{R}}^{n} \), then \( U \) is arcwise connected. | Proof. We may assume \( U \) to be nonempty. Let \( {x}_{0} \) be a point of \( U \), and denote by \( V \) the set of those points \( x \) in \( U \) such that there exists an arc in \( U \) joining \( {x}_{0} \) to \( x \) . If \( x \in V \) and if \( W \) is an open cell in \( {\mathbb{R}}^{n} \) containing \( x \) ... | Yes |
Proposition 3.9. Every open subset \( U \) of \( \mathbb{C} \) is uniquely expressible as a countable union of disjoint domains, and these domains are the components of \( U \) . | Proof. The empty set is the empty union of domains. If \( U \) is a nonempty open set in \( \mathbb{C} \), and if \( {\lambda }_{0} \) is an element of some component \( {U}_{0} \) of \( U \), then (Ex. A) there is an open disc \( {D}_{r}\left( {\lambda }_{0}\right) \) about \( {\lambda }_{0}\left( {r > 0}\right) \) su... | No |
Proposition 3.10. Let \( K \) be a compact subset of \( \mathbb{C} \), and suppose \( L \) is a compact subset of \( \mathbb{C} \) such that \( K \subset L \) and such that \( \partial L \subset K \) . Then \( L \) consists of the union of \( K \) and some of the holes of \( K \) . In particular, if \( K \) has no hole... | Proof. According to Proposition 3.6 each component of \( \mathbb{C} \smallsetminus K \) is either contained in \( L \) or disjoint from \( L \) . In particular, the unbounded component of \( \mathbb{C} \smallsetminus K \) is disjoint from \( L \) since \( L \) is bounded. | No |
Proposition 3.16. If \( X \) is a Hausdorff space, then no net in \( X \) converges to more than one point. | Proof. If \( x \) and \( y \) are distinct points of \( X \), then there are neighborhoods \( U \) and \( V \) of \( x \) and \( y \), respectively, such that \( U \) and \( V \) are disjoint. Now suppose that \( {\left\{ {z}_{\lambda }\right\} }_{\lambda \in \Lambda } \) is a net in \( X \) that converges to both \( x... | Yes |
Proposition 3.18. A mapping \( f \) of a topological space \( X \) into a topological space \( Y \) is continuous at a point \( {x}_{0} \) of \( X \) if and only if, for every net \( {\left\{ {x}_{\lambda }\right\} }_{\lambda \in \Lambda } \) in \( X \) converging to \( {x}_{0} \), the net \( {\left\{ f\left( {x}_{\lam... | Proof of Proposition 3.18. Suppose first that \( f \) is continuous at \( {x}_{0} \), and let \( \left\{ {x}_{\lambda }\right\} \) be a net in \( X \) that converges to \( {x}_{0} \). For any neighborhood \( V \) of \( f\left( {x}_{0}\right) \) in \( Y \), there exist a neighborhood \( W \) of \( {x}_{0} \) such that \... | Yes |
Corollary 3.21. If \( {\left\{ {f}_{\gamma }\right\} }_{\gamma \in \Gamma } \) is an indexed family of mappings \( {f}_{\gamma } : X \rightarrow {Y}_{\gamma } \) of a set \( X \) into topological spaces \( {Y}_{\gamma } \), and if \( g \) is a mapping of a topological space \( Z \) into \( X \), then \( g \) is continu... | Proof. It suffices to prove the first assertion of the corollary. The condition is clearly necessary since the mappings \( {f}_{\gamma } \) are all continuous with respect to the topology they induce. To verify the sufficiency of the stated criterion, suppose that all of the compositions \( {f}_{\gamma } \circ g \) are... | Yes |
Theorem 4.2. A normal topological space that satisfies the second axiom of countability is metrizable. | Proof. Let \( \mathcal{B} \) be a countable base for \( X \) . There are only countably many pairs \( \left( {U, V}\right) \) of sets in \( \mathcal{B} \) such that \( {V}^{ - } \subset U \) . Let \( {\left\{ \left( {U}_{n},{V}_{n}\right) \right\} }_{n = 1}^{\infty } \) be an enumeration of the set of all such pairs, a... | Yes |
Proposition 4.6. Every continuous mapping \( f \) of a compact metric space \( \left( {X,\rho }\right) \) into a metric space \( \left( {Y,{\rho }^{\prime }}\right) \) is uniformly continuous. | Proof. Let \( \varepsilon \) be a positive number. For each point \( x \) of \( X \) there exists a \( {\delta }_{x} > 0 \) such that \( {\rho }^{\prime }\left( {f\left( {x}^{\prime }\right), f\left( x\right) }\right) < \varepsilon /2 \) whenever \( {x}^{\prime } \) is a point of \( X \) such that \( \rho \left( {{x}^{... | Yes |
Proposition 4.7. A set \( N \) is nowhere dense in a metric space \( X \) if and only if, for every nonempty open set \( U \) in \( X \), there exists a nonempty open set \( V \) such that \( V \subset U \) and \( V \cap N = \varnothing \) . Moreover, if \( N \) is nowhere dense, it is always possible to arrange for \(... | Proof. If \( N \) is not nowhere dense, and \( U \) is a nonempty open set contained in \( {N}^{ - } \), then every nonempty open subset \( V \) of \( U \) clearly meets \( N \) . This proves the sufficiency of the given condition. To prove the necessity, set \( V = U \smallsetminus {N}^{ - } \) . If \( V = \varnothing... | Yes |
Theorem 4.8 (Baire Category Theorem). Let \( X \) be a complete metric space, and let \( U \) be a nonempty open subset of \( X \) . Then \( U \) is of second category in \( X \) . | Proof. Let \( {\left\{ {N}_{n}\right\} }_{n = 1}^{\infty } \) be an arbitrary sequence of nowhere dense sets in \( X \) . It suffices to show that if \( A = \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{N}_{n} \), then \( U \smallsetminus A \neq \varnothing \) . Note first that by the foregoing proposition there exists ... | Yes |
Theorem 5.2 (Identity Theorem). Let \( f \) and \( g \) be analytic functions on the same domain \( \Delta \), and suppose \( f\left( \lambda \right) = g\left( \lambda \right) \) for all \( \lambda \) in some subset \( M \) of \( \Delta \) possessing an accumulation point in \( \Delta \) . Then \( f \equiv g \) on \( \... | Proof. It clearly suffices to show that \( f = 0 \) identically on \( \Delta \) when \( f = 0 \) on \( M \) . Let \( U \) denote the subset of \( \Delta \) consisting of all those points \( \lambda \) in \( \Delta \) such that \( f \) vanishes identically on some open disc of positive radius about \( \lambda \) . Then ... | Yes |
Theorem 5.4. Let \( \Delta \) be a domain in \( \mathbb{C} \), let \( {\alpha }_{0} \) be a point in \( \Delta \), and let \( f \) be a function that is continuous on \( \Delta \) and analytic on \( \Delta \smallsetminus \left\{ {\alpha }_{0}\right\} \) . If \( \alpha ,\beta \), and \( \gamma \) are complex numbers suc... | For a proof of this fundamental result when \( f \) is assumed to be analytic on the entire domain \( \Delta \) we refer the reader to any standard textbook on complex analysis. For a proof of the theorem as stated here one may consult [57]. | No |
Proposition 5.5. Let \( f \) be a function defined and continuous on a domain \( \Delta \) and suppose \( f \) possesses a primitive \( F \) on \( \Delta \) . In other words, suppose there is an analytic function \( F \) on \( \Delta \) whose derivative is \( f \) . Then for an arbitrary rectifiable arc \( \alpha \) in... | Example E. If \( p\left( \lambda \right) = {\alpha }_{n}{\lambda }^{n} + \cdots + {\alpha }_{0} \) is a polynomial, and if \( \gamma \) is an arbitrary closed rectifiable arc in \( \mathbb{C} \), then \( {\int }_{\gamma }p\left( \lambda \right) {d\lambda } = 0 \), since \( p \) possesses the primitive\n\n\[ \n\frac{{\a... | No |
Theorem 5.6. Let \( \gamma \) be a closed rectifiable arc in \( \mathbb{C} \) and let \( W \) denote the range of \( \gamma \) . Then the function\n\n\[ \n{w}_{\gamma }\left( \lambda \right) = \frac{1}{2\pi i}{\int }_{\gamma }\frac{d\zeta }{\zeta - \lambda }\n\]\n\nis constant and integer-valued on each component of th... | Proof. Since \( W \) is compact it is easily seen that \( {w}_{\gamma }\left( \lambda \right) \) tends to zero as \( \lambda \) tends to infinity. Similarly it is easy to verify that \( {w}_{\gamma } \) is continuous on \( \mathbb{C} \smallsetminus W \) . (If \( {\lambda }_{0} \notin W \) and if \( \lambda \) is suffic... | Yes |
Theorem 5.7 (Cauchy-Goursat Theorem in a Disc). Let \( D = {D}_{R}\left( \alpha \right) \) be an open disc in \( \mathbb{C}\left( {R > 0}\right) \), let \( {\alpha }_{0} \) be a point in \( D \), and let \( f \) be a continuous function on \( D \) that is analytic on \( D \smallsetminus \left\{ {\alpha }_{0}\right\} \)... | Proof. For each point \( \lambda \) in \( D \) we define \( F\left( \lambda \right) = {\int }_{\sigma }f\left( \zeta \right) {d\zeta } \), where \( \sigma \) denotes the directed line segment \( \sigma \left( {\alpha ,\lambda }\right) \) . Then according to Theorem 5.4 we have \( F\left( \lambda \right) - F\left( {\lam... | Yes |
Theorem 5.8 (Cauchy Integral Formula in a Disc). Let \( D = {D}_{R}\left( \alpha \right) \) be a disc in \( \mathbb{C}\left( {R > 0}\right) \), let \( f \) be a function defined and analytic on \( D \), and let \( \gamma \) be a closed rectifiable arc in D. Then\n\n\[ \n{w}_{\gamma }\left( \lambda \right) f\left( \lamb... | Proof. The function\n\n\[ \ng\left( \zeta \right) = \left\{ \begin{array}{ll} \frac{f\left( \zeta \right) - f\left( \lambda \right) }{\zeta - \lambda }, & \zeta \neq \lambda \\ {f}^{\prime }\left( \lambda \right) , & \zeta = \lambda \end{array}\right.\n\]\n\nis continuous on \( D \) and analytic on \( D \smallsetminus ... | Yes |
Proposition 5.9. Let \( {r}_{0} \) and \( {r}_{1} \) be nonnegative numbers such that \( {r}_{0} < {r}_{1} \), let \( \alpha \) be a complex number, and suppose \( f \) is an analytic function on the annular domain \( A = \left\{ {\lambda \in \mathbb{C} : {r}_{0} < \left| {\lambda - \alpha }\right| < {r}_{1}}\right\} \... | The coefficients \( {\alpha }_{n} \) in (15) are uniquely determined by \( f \), and the convergence in (15) is absolute in \( A \) and uniform on compact subsets of \( A \). The series (15) is known as the Laurent expansion of \( f \) in \( A \). | Yes |
Proposition 5.10. Let \( f \) be a complex-valued function on a complex domain \( \Delta \) , and let \( u \) and \( v \) be the real and imaginary parts of \( f \) on the corresponding real domain \( {\Delta }^{\prime } \) . If \( f \) is analytic on \( \Delta \) then \( u \) and \( v \) satisfy the equations\n\n\[ \n... | Proof. Let \( {\lambda }_{0} = {x}_{0} + i{y}_{0} \) be a point of \( \Delta \left( {{x}_{0},{y}_{0}}\right. \) real \( ) \), let \( R \) be a positive radius small enough so that the disc \( {D}_{R}\left( {\lambda }_{0}\right) \subset \Delta \), and suppose that \( f \) is differentiable at \( {\lambda }_{0} \) . If \... | Yes |
Proposition 5.12. If \( u \) is a real harmonic function on a disc\n\n\[ \n{D}^{\prime } = \left\{ {\left( {x, y}\right) : {\left( x - {x}_{0}\right) }^{2} + {\left( y - {y}_{0}\right) }^{2} < {r}^{2}}\right\} \;\left( {r > 0}\right)\n\]\n\nin \( {\mathbb{R}}^{2} \), then \( u \) possesses a harmonic conjugate \( \wide... | Proof. The uniqueness of the harmonic conjugate follows from the fact that the Cauchy-Riemann equations must be satisfied (see also Problem B). To see that \( \widetilde{u} \) exists, we observe that for any function \( u \) in \( {\mathcal{C}}_{\mathbb{R}}^{\left( 1\right) }\left( {D}^{\prime }\right) \) the functions... | Yes |
Proposition 7.1. Let \( L \) be a Lebesgue integral on a measurable space \( X \) and let \( \mathcal{L} \) denote the linear space of functions integrable with respect to \( L \) . If \( f \) is a real-valued [nonnegative] function in \( \mathcal{L} \), then \( L\left( f\right) \) is also real [nonnegative]. Hence \( ... | Proof. If \( f \) is a nonnegative function that belongs to \( \mathcal{L} \), then \( f = \left| f\right| \) , and it follows that \( L\left( f\right) \geq 0 \) (set \( g = 0 \) in axiom \( \left( {\mathrm{L}}_{1}\right) \) ). If \( f \) is real-valued and belongs to \( \mathcal{L} \), then the positive and negative p... | Yes |
Proposition 7.2. Let L be a (real or complex) Lebesgue integral on a measurable space \( X \), and let \( \left\{ {f}_{n}\right\} \) be a monotone increasing sequence of nonnegative functions on \( X \), each of which is integrable with respect to \( L \) . Suppose also that \( \left\{ {f}_{n}\right\} \) converges poin... | Proof. According to Proposition 7.1 the sequence of integrals \( \left\{ {L\left( {f}_{n}\right) }\right\} \) is nonnegative and monotone increasing. Moreover, if \( g \) is integrable, then \( 0 \leq L\left( {f}_{n}\right) \leq L\left( g\right) \) for every \( n \) . On the other hand, if the sequence \( \left\{ {L\le... | Yes |
Theorem 7.6. Let \( \left( {X,\mathbf{S}}\right) \) be a measurable space, and suppose given a Lebesgue integral [real Lebesgue integral] \( \left( {X,\mathcal{L}, L}\right) \) on \( X \) . For each measurable set \( E \) in \( X \), define\n\n\[ \n\mu \left( E\right) = \left\{ \begin{array}{ll} L\left( {\chi }_{E}\rig... | The proof that (6) defines a measure on \( \left( {X,\mathbf{S}}\right) \) is an easy exercise (Prob. A). Furthermore, the uniqueness of the Lebesgue integral [real Lebesgue integral] \( L \) satisfying (6) with respect to a given measure \( \mu \) is a consequence of the fact that (6) determines which characteristic f... | No |
Let \( \left( {X,\mathbf{S},\mu }\right) \) be a measure space. If \( f \) is a nonnegative measurable function on \( X \), then \( \int {fd\mu } = 0 \) if and only if \( f = 0 \) a.e. \( \left\lbrack \mu \right\rbrack \). | Proof. Let \( f \) be a nonnegative measurable function on \( X \), and let \( \left\{ {s}_{n}\right\} \) be a monotone increasing sequence of nonnegative measurable simple functions converging pointwise to \( f \) (Prop. 6.6). If \( \int {fd\mu } = 0 \), then \( \int {s}_{n}{d\mu } = 0 \) and therefore \( {s}_{n} = 0 ... | Yes |
Corollary 7.10 (Theorem of Beppo-Levi). Let \( \left( {X,\mathbf{S},\mu }\right) \) be a measure space, let \( \left\{ {p}_{n}\right\} \) be a sequence of extended real-valued functions defined and nonnegative a.e. \( \left\lbrack \mu \right\rbrack \) and measurable \( \left\lbrack \mu \right\rbrack \) on \( X \), and ... | Proof. Just as in the preceding proof it is easy to see that it is enough to treat the case in which the functions \( {p}_{n} \) are defined and nonnegative everywhere on \( X \) . Set \( {f}_{n} = {p}_{1} + {p}_{2} + \cdots + {p}_{n}, n = 1,2,\ldots \), and apply the monotone convergence theorem. | Yes |
Theorem 7.11 (Fatou’s Lemma). Let \( \left( {X,\mathbf{S},\mu }\right) \) be a measure space, and let \( \left\{ {f}_{n}\right\} \) be a sequence of functions defined and nonnegative a.e. \( \left\lbrack \mu \right\rbrack \) and integrable \( \left\lbrack \mu \right\rbrack \) on \( X \) . Suppose that there exists a re... | Proof. For each positive integer \( n \) the function \( {g}_{n} = \mathop{\inf }\limits_{{k \geq n}}{f}_{k} \) is defined and nonnegative almost everywhere. Moreover, \( {g}_{n} \) is measurable and, since \( 0 \leq {g}_{n} \leq {f}_{n} \) a.e., it follows that \( {g}_{n} \) is integrable \( \left\lbrack \mu \right\rb... | Yes |
Theorem 7.12 (Dominated Convergence Theorem). Let \( \left( {X,\mathbf{S},\mu }\right) \) be a measure space, let \( \left\{ {f}_{n}\right\} \) be a sequence of complex-valued functions measurable \( \left\lbrack \mu \right\rbrack \) on \( X \), and suppose that \( \left\{ {f}_{n}\right\} \) converges a.e. to a functio... | \[ \mathop{\lim }\limits_{n}{\int }_{X}\left| {{f}_{n} - f}\right| {d\mu } = 0 \] and therefore \[ {\int }_{X}{fd\mu } = \mathop{\lim }\limits_{n}{\int }_{X}{f}_{n}{d\mu } \] | Yes |
Theorem 7.13 (Bounded Convergence Theorem). Let \( \left( {X,\mathbf{S},\mu }\right) \) be a finite measure space, and let \( \left\{ {f}_{n}\right\} \) be a sequence of complex-valued functions measurable \( \left\lbrack \mu \right\rbrack \) on \( X \) . Suppose that \( \left\{ {f}_{n}\right\} \) converges a.e. to a f... | \[ \mathop{\lim }\limits_{n}{\int }_{X}\left| {{f}_{n} - f}\right| {d\mu } = 0 \] and therefore \[ {\int }_{X}{fd\mu } = \mathop{\lim }\limits_{n}{\int }_{X}{f}_{n}{d\mu } \] The bounded convergence theorem is an easy consequence of the dominated convergence theorem, which can, in turn, be derived from the monotone con... | No |
A necessary and sufficient condition for a measure space \( \left( {X,\mathbf{S},\mu }\right) \) to be locally finite is that it possess no infinite atoms. | Proof. If \( \mu \left( A\right) = + \infty \) and \( A \) is an atom with respect to \( \mu \), then \( \mu \left( F\right) < + \infty \) and \( F \subset A \) imply \( \mu \left( F\right) = 0 \), so \( \mu \) is certainly not locally finite. Suppose, in the converse direction, that \( \mu \) fails to be locally finit... | Yes |
Let \( {\left\{ \left( {X}_{\gamma },{\mathbf{S}}_{\gamma },{\mu }_{\gamma }\right) \right\} }_{\gamma \in \Gamma } \) be an indexed family of measure spaces with the property that the sets \( X \), are pairwise disjoint, and let \( \left( {X,\mathbf{S},\mu }\right) \) denote the direct sum of the given family. Then \(... | \[ {\int }_{X}{fd\mu } = \mathop{\sum }\limits_{{\gamma \in \Gamma }}\left( {{\int }_{{X}_{\gamma }}{fd}{\mu }_{\gamma }}\right) \] | No |
Lemma 8.4. If \( \mu \) is a signed measure on a measurable space \( \left( {X,\mathbf{S}}\right) \), and if \( \mu \left( E\right) < + \infty \) for some measurable set \( E \), then \( \mu \left( F\right) < + \infty \) for every measurable set \( F \) contained in \( E \) . Similarly, if \( \mu \left( E\right) > - \i... | Completion of Proof. Suppose \( \mu \left( E\right) = + \infty \) and \( \mu \left( F\right) = - \infty \) . Then \( - \infty < \mu \left( {E \cap F}\right) < + \infty \) by what has already been said, so \( \mu \left( {E \smallsetminus F}\right) = + \infty \) and \( \mu \left( {F \smallsetminus E}\right) = - \infty \)... | Yes |
Theorem 8.6. The positive and negative variations \( {\mu }_{ + } \) and \( {\mu }_{ - } \) of a signed measure \( \mu \) on a measurable space \( \left( {X,\mathbf{S}}\right) \) are measures on \( \left( {X,\mathbf{S}}\right) \), one of which at least is finite, and \( \mu = {\mu }_{ + } - {\mu }_{ - } \) setwise on \... | Proof. It is clear that \( {\mu }_{ + } \) and \( {\mu }_{ - } \) are measures, one of which is necessarily finite, that both are finite (or \( \sigma \) -finite) if \( \mu \) is, and that \( \mu = {\mu }_{ + } - {\mu }_{ - } \) . Suppose \( {v}_{1} \) and \( {v}_{2} \) are measures on \( \left( {X,\mathbf{S}}\right) \... | Yes |
Proposition 8.7. Let \( \zeta \) be a complex measure on a measurable space \( \left( {X,\mathbf{S}}\right) \), and let \( \mu \) and \( v \) denote the real and imaginary parts of \( \zeta \), respectively. Then \( \left| \mu \right| = {\mu }_{ + } + {\mu }_{ - },\left| v\right| = {v}_{ + } + {v}_{ - } \), and \( \lef... | \[ \left| \mu \right| \left( E\right) \vee \left| v\right| \left( E\right) \leq \left| \zeta \right| \left( E\right) \leq \left| \mu \right| \left( E\right) + \left| v\right| \left( E\right) ,\;E \in \mathbf{S}. \] | Yes |
Proposition 8.8. Let \( \zeta \) be a complex measure on a measurable space \( \left( {X,\mathbf{S}}\right) \) , and let \( \mathcal{L} \) denote the set of all those complex-valued measurable functions on \( X \) that are integrable \( \left\lbrack \zeta \right\rbrack \) . Then \( \mathcal{L} \) is a linear space of f... | Proof. It is obvious that \( \mathcal{L} \) is a linear space and that integration with respect to \( \zeta \) is a linear functional on \( \mathcal{L} \) . In order to verify (8) we note that this inequality is clearly valid when \( f \) is a simple function. Indeed, if \( s = \mathop{\sum }\limits_{{i = 1}}^{n}{\alph... | Yes |
Proposition 8.9. Let \( f \) be a continuous real-valued function defined on a closed interval \( \left\lbrack {a, b}\right\rbrack \), let \( U \) be an open subset of \( \left\lbrack {a, b}\right\rbrack \), and let \( m \) be a real number. Then the set \( {R}_{m}\left( {f;U}\right) \left\lbrack {{L}_{m}\left( {f;U}\r... | Proof. It clearly suffices to treat the case in which \( U \) is a single open interval, which we may as well assume to be \( \left( {a, b}\right) \) . Let \( g\left( t\right) = f\left( t\right) - {mt}, a \leq t \leq b \) . Then the difference quotient of \( g \) across any subinterval of \( \left\lbrack {a, b}\right\r... | Yes |
Lemma 8.10 (Riesz [54]). For any continuous real-valued function \( f \) defined on a closed interval \( \left\lbrack {a, b}\right\rbrack \), the set \( {R}_{0} = {R}_{0}\left( {f;\left( {a, b}\right) }\right) \) is open, and if \( \left( {c, d}\right) \) is any one of the components of \( {R}_{0} \), then \( f\left( c... | Proof. A point \( t \) of \( \left( {a, b}\right) \) belongs to \( {R}_{0} \) if and only if there exists a point \( u \) in \( \left( {t, b}\right) \) such that \( f\left( t\right) < f\left( u\right) \) . From this and the fact that \( f \) is continuous it is obvious that \( {R}_{0} \) is open. Let \( \left( {c, d}\r... | Yes |
Lemma 8.11. Let \( f \) be a continuous monotone increasing function defined on an interval \( \left\lbrack {a, b}\right\rbrack \) and let \( m \) and \( M \) be positive numbers such that \( m < M \) . Let\n\n\[ E = \left\{ {t \in \left( {a, b}\right) : {D}_{ + }\left( t\right) > M\;\text{ and }\;{d}_{ - }\left( t\rig... | Proof. Let \( V = {L}_{m}\left( {f;U}\right) \), and set \( W = {R}_{M}\left( {f;V}\right) \) . According to the above remarks we have \( E \subset W \subset V \subset U \) . Moreover, if \( \left( {c, d}\right) \) denotes any one component of \( V \), then, according to Proposition 8.9, \( f\left( d\right) - f\left( c... | Yes |
Theorem 8.13 (Lebesgue). Let \( f \) be a continuous monotone increasing function defined on an interval \( \left\lbrack {a, b}\right\rbrack \), and let \( {f}^{\prime } \) denote the derivative of \( f \) (existent a.e. \( \left\lbrack {\mu }_{1}\right\rbrack \) by the preceding theorem). Then \( {f}^{\prime } \) is L... | Proof. Let \( {\left\{ {\mathcal{P}}_{n}\right\} }_{n = 1}^{\infty } \) be a nested sequence of partitions of the interval \( \left\lbrack {a, b}\right\rbrack \) with the property that mesh \( {\mathcal{P}}_{n} \downarrow 0 \) (see Problem \( 1\mathrm{G} \) for basic definitions), let \( {\mathcal{P}}_{n} = \left\{ {a ... | No |
Proposition 8.14. If \( g \) is any continuous complex-valued function of bounded variation defined on an interval \( \left\lbrack {a, b}\right\rbrack \), then the derivative \( {g}^{\prime } \) exists a.e. \( \left\lbrack {\mu }_{1}\right\rbrack \) and satisfies the condition\n\n\[{\int }_{a}^{b}\left| {g}^{\prime }\r... | Proof. That the derivative of \( g \) exists a.e. \( \left\lbrack {\mu }_{1}\right\rbrack \) and is Lebesgue integrable on \( \left\lbrack {a, b}\right\rbrack \) follows at once from consideration of Jordan decompositions of the real and imaginary parts of \( g \) (Prob. 1K). To verify (9), let \( {\left\{ {\mathcal{P}... | Yes |
Theorem 9.2 (Fubini Theorem). If \( \left( {X,\mathbf{S},\mu }\right) \) and \( \left( {Y,\mathbf{T}, v}\right) \) are \( \sigma \) -finite measure spaces, then there exists a unique measure \( \mu \times v \) (called the product of \( \mu \) and \( v \) ) on the measurable space \( \left( {X \times Y,\mathbf{S} \times... | \[ {\int }_{X}{gd\mu } = {\int }_{X}\left\lbrack {{\int }_{Y}f\left( {x, y}\right) {dv}\left( y\right) }\right\rbrack {d\mu }\left( x\right) = {\int }_{X \times Y}{fd}\left( {\mu \times v}\right) . \] Similarly, the iterated integral \[ {\int }_{Y}\left\lbrack {{\int }_{X}f\left( {x, y}\right) {d\mu }\left( x\right) }\... | Yes |
Proposition 9.7. Let \( \lambda ,\mu \), and \( v \) be \( \sigma \) -finite measures on a measurable space \( \left( {X,\mathbf{S}}\right) \), and suppose that \( \lambda \ll \mu \ll v \). Then \[ \frac{d\lambda }{dv} = \left( \frac{d\lambda }{d\mu }\right) \left( \frac{d\mu }{dv}\right) \] almost everywhere with resp... | Proof. If \( E \) is a set belonging to \( \mathbf{S} \), then \[ \lambda \left( E\right) = {\int }_{E}\left( \frac{d\lambda }{d\mu }\right) {d\mu } = {\int }_{E}\left( \frac{d\lambda }{d\mu }\right) \left( \frac{d\mu }{dv}\right) {dv}. \] | Yes |
Proposition 9.9. The function \( \dot{\mu } \) on the metric space \( \left( {{\dot{\mathbf{S}}}_{\mathcal{F}},\rho }\right) \) associated with an arbitrary measure space \( \left( {X,\mathbf{S},\mu }\right) \) satisfies the condition\n\n\[ \left| {\dot{\mu }\left( \left\lbrack E\right\rbrack \right) - \dot{\mu }\left(... | Proof. For any two sets \( E \) and \( F \) of finite measure in \( X \) it is easy to verify that\n\n\[ \left| {\mu \left( E\right) - \mu \left( F\right) }\right| \leq \mu \left( {E\nabla F}\right) \]\n\nwhence the first part of the proposition follows at once. Similarly, if \( \Phi \) denotes any one of the four Bool... | Yes |
Theorem 9.11. Let \( \Phi \) be a weak measure ring isomorphism of the measure ring \( \left( {\dot{\mathbf{S}},\dot{\mu }}\right) \) of a \( \sigma \) -finite measure space \( \left( {X,\mathbf{S},\mu }\right) \) onto the measure ring \( \left( {\dot{\mathbf{T}},\dot{v}}\right) \) of a \( \sigma \) -finite measure spa... | Sketch of Proof. If we define \( {v}_{0}\left( F\right) = \dot{\mu }\left( {{\Phi }^{-1}\left( \left\lbrack F\right\rbrack \right) }\right), F \in \mathbf{T} \), it is easy to verify that \( {v}_{0} \) is a \( \sigma \) -finite measure on \( \left( {Y,\mathbf{T}}\right) \) such that \( {v}_{0} \) is equivalent to \( v ... | Yes |
Proposition 10.1. If \( X \) is a \( \sigma \) -compact Hausdorff space, then the collection \( \mathcal{K} \) of all compact subsets of \( X \) generates the \( \sigma \) -ring \( \mathbf{B} \) of Borel subsets of \( X \) . | Proof. It suffices to show that every closed set \( F \) in \( X \) belongs to \( \mathbf{S}\left( \mathcal{K}\right) \) . If \( X = \left( {\mathop{\bigcup }\limits_{{n = 1}}^{\infty }{K}_{n}}\right. \), where each \( {K}_{n} \) is compact, then \( F = \left( {\mathop{\bigcup }\limits_{{n = 1}}^{\infty }\left( {F \cap... | Yes |
Proposition 10.2. If \( X \) is a \( \sigma \) -compact, locally compact, Hausdorff space, then the collection of all topologically bounded open sets in \( X \) generates the \( \sigma \) -ring B of Borel sets in \( X \) . | Proof. It suffices to show that every open subset \( U \) of \( X \) is a countable union of topologically bounded open subsets of \( X \) . We write \( X = \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{K}_{n} \), where each \( {K}_{n} \) is compact. According to Problem A, there exists for each positive integer \( n \)... | Yes |
Corollary 10.4. Let \( \mu \) be a Borel measure on a locally compact Hausdorff space \( X \), and let \( \mathcal{K} \) denote the collection of all compact subsets of \( X \) . If every set \( K \) in \( \mathcal{K} \) is outer regular with respect to \( \mu \), or if every topologically bounded open set in \( X \) i... | Proof. It is an immediate consequence of Problem A that the \( \sigma \) -ring generated by the collection of topologically bounded open sets coincides with \( \mathbf{S}\left( \mathcal{K}\right) \) . If \( X \) is \( \sigma \) -compact, this \( \sigma \) -ring coincides with the \( \sigma \) -ring of all Borel subsets... | No |
Corollary 10.5. If \( X \) is a locally compact Hausdorff space, and if \( {\mathcal{K}}_{0} \) denotes the collection of all compact \( {G}_{\delta } \) ’s in \( X \), then the sets belonging to the \( \sigma \) -ring \( \mathrm{S}\left( {\mathcal{K}}_{0}\right) \) are all regular with respect to every Borel measure o... | Proof. Every set \( {K}_{0} \) in \( {\mathcal{K}}_{0} \) can be written as the intersection of a decreasing sequence of open sets, each of which may be taken to be topologically bounded (Prob. A). Hence \( {K}_{0} \) is outer regular by Problem 7F. | No |
If \( {X}^{ \cdot } \) is a locally compact Hausdorff space that is metrizable, then the sets belonging to the \( \sigma \) -ring \( \mathbf{S}\left( \mathcal{K}\right) \) generated by the collection \( \mathcal{K} \) of compact sets in \( X \) are all regular with respect to every Borel measure on \( X \) . If \( X \)... | Proof. If \( X \) is metrizable, then every compact set in \( X \) is a \( {G}_{\delta } \) (Prob. 6M). If \( X \) is a locally compact Hausdorff space satisfying the second axiom of countability, then \( X \) is metrizable and \( \sigma \) -compact (Prob. B). | No |
Proposition 10.7. Let \( \mu \) be a regular Borel measure on a locally compact Hausdorff space \( X \) . Then for any Borel measurable function \( f \) on \( X \) that is integrable \( \left\lbrack \mu \right\rbrack \) there exists a sequence \( \left\{ {g}_{n}\right\} \) of integrable continuous functions on \( X \) ... | Proof. It suffices to show that if \( f \) is integrable \( \left\lbrack \mu \right\rbrack \), then there exists a continuous function \( g \) on \( X \) such that \( \int \left| {f - g}\right| {d\mu } \) is as small as desired. To this end, suppose first that \( E \) is a Borel set such that \( \mu \left( E\right) < +... | Yes |
Proposition 10.8. Let \( \\mu \) and \( v \) be regular Borel measures on the same locally compact Hausdorff space \( X \), and suppose the integrals with respect to \( \\mu \) and \( v \) agree on continuous functions; that is, suppose \( {\\mathcal{L}}_{\\mu } \) and \( {\\mathcal{L}}_{v} \) contain the same set \( \... | Proof. Let \( K \) be a compact set in \( X \), and let\n\n\[ \n{U}_{1} \\supset {U}_{2} \\supset \\cdots \\supset {U}_{n} \\supset \\cdots \\supset K\n\]\n\nbe a nested sequence of topologically bounded open sets such that \( \\mu \\left( {U}_{n}\\right) \\rightarrow \\mu \\left( K\\right) \) and \( v\\left( {U}_{n}\\... | Yes |
Lemma 10.10. Let \( {\varphi }_{0} \) be a positive linear functional on \( \mathcal{C}\left( X\right) \), where \( X \) is a compact Hausdorff space, and suppose in addition that \( {\varphi }_{0}\left( 1\right) = 1 \) . For each open set \( U \) in \( X \) let\n\n\[ \rho \left( U\right) = \sup \left\{ {{\varphi }_{0}... | Proof. Clearly \( \rho \left( \varnothing \right) = 0 \) . Let \( V \) be an open subset of \( X \), let \( \left\{ {U}_{n}\right\} \) be a countable open covering of \( V \), let \( \varepsilon \) be a positive number, and let \( f \) be a continuous function on \( X \) such that \( 0 \leq f \leq {\chi }_{V} \) . We s... | Yes |
Lemma 10.11. Let \( X,{\varphi }_{0} \) and \( \rho \) be as in the preceding lemma, let \( U \) be an open set in \( X \), and let \( \varepsilon \) be a positive number. If \( f \) is a continuous function on \( X \) such that \( 0 \leq f \leq {\chi }_{U} \) and \( {\varphi }_{0}\left( f\right) > \rho \left( U\right)... | Proof. Suppose \( g \) is a continuous function on \( X \) such that \( 0 \leq g \leq {\chi }_{U \smallsetminus K} \) . Then \( f + g \leq {\chi }_{U} + \varepsilon \) . If we set \( h = \left( {f + g - \varepsilon }\right) \vee 0 \), then \( 0 \leq h \leq {\chi }_{U} \) and \( f + g \leq h + \varepsilon \), so we have... | Yes |
Lemma 10.12. Let \( X,{\varphi }_{0} \) and \( \rho \) be as in Lemma 10.10, and let \( U \) and \( V \) be open sets in \( X \) . Then\n\n\[ \rho \left( U\right) + \rho \left( V\right) \leq \rho \left( {U \cup V}\right) + \rho \left( {U \cap V}\right) \] | Proof. Let \( f \) and \( g \) be arbitrary continuous functions on \( X \) such that \( 0 \leq f \leq {\chi }_{U} \) and \( 0 \leq g \leq {\chi }_{V} \), and set\n\n\[ h = \left( {f + g}\right) \land 1\text{ and }k = f + g - h. \]\n\nThen it is easy to verify that \( 0 \leq h \leq {\chi }_{U \cup V} \) and \( 0 \leq k... | Yes |
Lemma 10.13. Let \( X,{\varphi }_{0} \) and \( \rho \) be as in Lemma 10.10, and let \( {\mu }^{ * } \) denote the outer measure on \( X \) generated by \( \rho \) (see Problem 8D for definitions). Then \( {\mu }^{ * } \) agrees with \( \rho \) on open sets, and\n\n\[{\mu }^{ * }\left( {V \smallsetminus U}\right) + {\m... | Proof. That \( {\mu }^{ * } \) agrees with \( \rho \) on open sets follows at once from Lemma 10.10. Let \( U \) and \( V \) be open sets, and let \( \varepsilon \) be any positive number. Let \( f \) be any continuous function on \( X \) such that \( 0 \leq f \leq {\chi }_{U} \) and \( {\varphi }_{0}\left( f\right) > ... | Yes |
Proposition 10.14. Let \( X \) be a locally compact Hausdorff space. If \( \mu \) is a signed Borel measure on \( X \), then \( {\mu }_{ + },{\mu }_{ - } \) and \( \left| \mu \right| \) are Borel measures on \( X \) . If \( \xi \) is a complex Borel measure on \( X \), then \( \operatorname{Re}\xi \) and \( \operatorna... | Proof. If \( \mu \) is a signed Borel measure on \( X \), and if \( K \) is a compact subset of \( X \), then both \( {\mu }_{ + }\left( K\right) \) and \( {\mu }_{ - }\left( K\right) \) must be finite, since \( \mu \left( K\right) = {\mu }_{ + }\left( K\right) - {\mu }_{ - }\left( K\right) \) is. But then \( \left| \m... | Yes |
Proposition 10.15. A signed Borel measure \( \mu \) on a locally compact Hausdorff space \( X \) is regular if and only if its positive and negative variations \( {\mu }_{ + } \) and \( {\mu }_{ - } \) are both regular. A complex Borel measure \( \xi \) on \( X \) is regular if and only if \( \operatorname{Re}\xi \) an... | Thus a regular complex Borel measure is simply a linear combination of (four) ordinary regular Borel measures. It should be noted that if \( \xi \) is a regular complex Borel measure on a locally compact Hausdorff space \( X \), if \( E \) is a Borel set in \( X \), and if \( \varepsilon \) is a positive number, then t... | No |
Proposition 11.1. If \( \rho \) is an invariant metric on a (real or complex) linear space \( \mathcal{E} \) , then\n\n\[ v\left( x\right) = \rho \left( {x,0}\right) \]\n\n(1)\ndefines a value on \( \mathcal{E} \), and the given metric \( \rho \) is the metric defined by \( v \) . | Proof. Suppose \( \rho \) is a given invariant metric on \( \mathcal{E} \), and let \( v \) be defined as in (1). Then \( v\left( x\right) > 0 \) for all \( x \neq 0 \), and \( v\left( {-x}\right) = \rho \left( {-x,0}\right) = \rho \left( {0, x}\right) = \rho \left( {x,0}\right) = \) \( v\left( x\right) \) for all \( x... | Yes |
Proposition 11.2. Let \( \mathcal{E} \) be a real or complex linear space and let \( v \) be a value on \( \mathcal{E} \) . Then \( \left( {x, y}\right) \rightarrow x + y \) is a continuous \( \mathcal{E} \) -valued mapping on \( \mathcal{E} \times \mathcal{E} \) . Likewise, the real-valued function \( x \rightarrow v\... | Proof. The proposition results immediately from the following inequalities, which are themselves easy consequences of the defining properties of a value:\n\n\[ \rho \left( {{x}_{1} + {y}_{1},{x}_{2} + {y}_{2}}\right) = v\left( {\left( {{x}_{1} + {y}_{1}}\right) - \left( {{x}_{2} + {y}_{2}}\right) }\right) \leq v\left( ... | Yes |
Proposition 11.4. If \( \mathcal{E} \) is a normed space, then the metric \( \rho \) defined on \( \mathcal{E} \) by the norm is an invariant metric satisfying the condition\n\n\[ \rho \left( {{\lambda x},{\lambda y}}\right) = \left| \lambda \right| \rho \left( {x, y}\right) ,\;x, y \in \mathcal{E},\;\lambda \in \mathb... | Proof. One way is clear enough; if \( \rho \) is defined by a norm \( \parallel \parallel \) on \( \mathcal{E} \), then \( \rho \left( {{\lambda x},{\lambda y}}\right) = \parallel {\lambda x} - {\lambda y}\parallel = \left| \lambda \right| \parallel x - y\parallel = \left| \lambda \right| \rho \left( {x, y}\right) \) f... | Yes |
Proposition 11.5. If \( \mathcal{E} \) is a normed space with norm \( \parallel \parallel \), then the \( \mathcal{E} \) -valued mappings \( \left( {\alpha, x}\right) \rightarrow {\alpha x} \) and \( \left( {x, y}\right) \rightarrow x + y \) are continuous. Likewise, the real function \( x \rightarrow \parallel x\paral... | Proof. The continuity of the mapping \( \left( {x, y}\right) \rightarrow x + y \) and the facts concerning the function \( x \rightarrow \parallel x\parallel \) have already been established, more generally, for vector spaces equipped with a value (Prop. 11.2), and these assertions are included here purely for convenie... | No |
Proposition 11.7. The intersection of an arbitrary nonempty collection of subspaces of a normed space \( \mathcal{E} \) is again a subspace of \( \mathcal{E} \). Consequently, given an arbitrary subset \( M \) of \( \mathcal{E} \), there exists a smallest subspace \( \mathcal{M} \) of \( \mathcal{E} \) that contains \(... | Proof. An intersection of linear manifolds in a linear space is a linear manifold, and an intersection of closed sets in a topological space is a closed set. Hence an intersection of subspaces of \( \mathcal{E} \) is a subspace of \( \mathcal{E} \). The smallest subspace of \( \mathcal{E} \) containing a given subset \... | Yes |
Proposition 11.9. The topological closure \( {\mathcal{L}}^{ - } \) of a linear manifold \( \mathcal{L} \) in a normed space \( \mathcal{E} \) is again a linear manifold, and is therefore a subspace of \( \mathcal{E} \) . Hence the subspace spanned by a subset \( M \) of \( \mathcal{E} \) coincides with the closure of ... | Proof. The second and third assertions are immediate consequences of the first. To prove the first, let \( x \) and \( y \) belong to \( {\mathcal{L}}^{ - } \) and let \( \left\{ {x}_{n}\right\} \) and \( \left\{ {y}_{n}\right\} \) be sequences in \( \mathcal{L} \) such that \( {x}_{n} \rightarrow x \) and \( {y}_{n} \... | Yes |
Theorem 11.10. If E is a Banach space and \( \mathcal{M} \) is a subspace of E, then \( \mathcal{E}/\mathcal{M} \) is also a Banach space. | Proof. We must show that \( \mathcal{E}/\mathcal{M} \) is complete. Accordingly, we suppose given a Cauchy sequence \( {\left\{ \left\lbrack {x}_{n}\right\rbrack \right\} }_{n = 1}^{\infty } \) of cosets in \( \mathcal{E}/\mathcal{M} \) . As noted (Prob. H) the natural projection of \( \mathcal{E} \) onto \( \mathcal{E... | Yes |
Proposition 11.11. The direct sum \( {\mathcal{E}}_{1}{ \oplus }_{1}\cdots { \oplus }_{1}{\mathcal{E}}_{n} \) of a finite sequence \( \left\{ {{\mathcal{E}}_{1},\ldots ,{\mathcal{E}}_{n}}\right\} \) of normed spaces is complete if and only if each space \( {\mathcal{E}}_{i} \) is complete, \( i = 1,\ldots, n \) . | Proof. A sequence \( {\left\{ \left( {x}_{1}^{\left( k\right) },\ldots ,{x}_{n}^{\left( k\right) }\right) \right\} }_{k = 1}^{\infty } \) is Cauchy [convergent] in \( {\mathcal{E}}_{1}{ \oplus }_{1}\cdots { \oplus }_{1}{\mathcal{E}}_{n} \) if and only if each sequence \( {\left\{ {x}_{i}^{\left( k\right) }\right\} }_{k... | Yes |
Lemma 11.13. If \( \mathcal{E} \) is a quasinormed linear space and \( \alpha \) is a fixed complex number, then the dilatation \( x \rightarrow {\alpha x} \) is a continuous mapping of \( \mathcal{E} \) into itself. | Proof. It suffices to show that if \( \varepsilon \) is a given positive number, then there exists a positive number \( \delta \) such that \( \parallel x\parallel < \delta \) implies \( \parallel {\alpha x}\parallel < \varepsilon \) . Moreover, for given \( \varepsilon > 0 \) we know that there exists \( \delta > 0 \)... | Yes |
Proposition 11.14. If \( \mathcal{E} \) is a quasinormed linear space with quasinorm ||, then the \( \mathcal{E} \) -valued mappings \( \left( {\alpha, x}\right) \rightarrow {\alpha x} \) and \( \left( {x, y}\right) \rightarrow x + y \) are continuous. Likewise, the real function \( x \rightarrow \parallel x\parallel \... | Proof. Since I I is a value; the continuity of the mapping \( \left( {x, y}\right) \rightarrow x + y \) and the facts concerning the function I have already been established (Prop. 11.2), and these assertions are included here purely for convenience of reference. In order to see that the mapping \( \left( {\alpha, x}\r... | Yes |
Theorem 11.16. If \( \mathcal{E} \) is a quasinormed space, then there exists an \( F \) -space \( \widehat{\mathcal{E}} \) and a linear isomorphism \( \varphi \) of \( \mathcal{E} \) onto a dense linear manifold \( \varphi \left( \mathcal{E}\right) \) in \( \widehat{\mathcal{E}} \) such that \( \parallel \varphi \left... | Proof. Suppose first that \( \left( {\widehat{\mathcal{E}},\varphi }\right) \) and \( \left( {{\widehat{\mathcal{E}}}_{1},{\varphi }_{1}}\right) \) are two pairs satisfying the stated conditions. Setting \( {\Phi }_{0}\left( {\varphi \left( x\right) }\right) = {\varphi }_{1}\left( x\right), x \in \mathcal{E} \), define... | No |
Proposition 11.17. Let \( \sigma \) be a pseudonorm on a linear space \( \mathcal{E} \), and let \( \mathcal{L} \) denote the zero space with respect to \( \sigma \) . Then, in the topology induced on \( \mathcal{E} \) by \( \sigma ,\mathcal{Y} \) is a closed linear manifold in \( \mathcal{E} \) coinciding with the clo... | Proof. That \( \mathcal{L} \) is a linear manifold in \( \mathcal{E} \) is clear from the defining properties of a pseudonorm. If \( x \notin \mathcal{Z} \), then \( d = \sigma \left( x\right) > 0 \), and if \( \sigma \left( {x - y}\right) < d/2 \), then \( \sigma \left( y\right) > d/2 \) by the triangle inequality. Th... | No |
Corollary 11.19. The weight of a topological linear space (Prob. 3A) is the same at every one of its points. In particular, a topological linear space . satisfies the first axiom of a countability if and only if there exists a countable neighborhood base at the origin 0 in 6 . | Proof. According to Proposition 11.18, if \( {x}_{0} \) is any vector in \( \mathcal{E} \), then \( \mathcal{V} \) is a neighborhood base at the origin 0 if and only if \( {x}_{0} + \mathcal{V} = \left\{ {{x}_{0} + V : V \in \mathcal{V}}\right\} \) is a neighborhood base at \( {x}_{0} \) . | Yes |
Proposition 11.20. The following conditions are equivalent for an arbitrary topological linear space \( \mathcal{E} \) :\n\n(i) \( \mathcal{E} \) is separated,\n\n(ii) Every singleton in \( \mathcal{E} \) is a closed set,\n\n(iii) At least one singleton in \( \mathcal{E} \) is a closed set,\n\n(iv) The origin 0 is not ... | Proof. It is clear that (i) implies (ii) and that (ii) implies (iv). Moreover, Proposition 11.18 shows that (ii) and (iii) are equivalent in every topological linear space. To complete the proof, suppose (iv) holds and let \( x \) be a vector in \( \mathcal{E} \) such that \( x \neq 0 \) . Then there exists a neighborh... | Yes |
Proposition 11.21. If \( \mathcal{L} \) is a linear manifold in a topological linear space \( \mathcal{E} \) , then \( {\mathcal{L}}^{ - } \) is also a linear manifold in \( \mathcal{E} \) . | Proof. Let \( x \) and \( y \) be vectors in \( {\mathcal{L}}^{ - } \) and suppose \( V \) is a neighborhood of \( x + y \) . There exist neighborhoods \( {W}_{1} \) and \( {W}_{2} \) of \( x \) and \( y \), respectively, such that \( {W}_{1} + {W}_{2} \subset V \) . Since both \( {W}_{1} \) and \( {W}_{2} \) contain v... | Yes |
Proposition 11.22. Let \( \mathcal{E} \) be a topological linear space and let \( \mathcal{M} \) be a linear submanifold of E. Then the quotient topology on E/M (Prob. 3H) is a linear topology with the property that the natural projection \( \pi \) of \( \mathcal{E} \) onto \( \mathcal{E}/\mathcal{M} \) is both open an... | Proof. It is clear in general from the definition that the natural projection \( \pi \) is continuous. (Recall that a set \( U \) in \( \mathcal{E}/\mathcal{M} \) is only said to be open in the quotient topology when \( {\pi }^{-1}\left( U\right) \) is open in \( \mathcal{E} \) .) In the case at hand, if \( A \) is an ... | Yes |
Proposition 11.25. If \( \mathcal{E} \) is a linear space and \( {\left\{ {\mathcal{T}}_{\gamma }\right\} }_{\gamma \in \Gamma } \) is a family of linear topologies on \( \mathcal{E} \), then the topology \( \mathcal{T} = \mathop{\sup }\limits_{\gamma }{\mathcal{T}}_{\gamma }\left( {\text{Prob.}{3G}}\right) \) is also ... | Proof. Let \( x \) and \( y \) be vectors in \( \mathcal{E} \), and let\n\n\[ U = {U}_{1} \cap \cdots \cap {U}_{n} \]\n\nbe a typical basic open set in \( \mathcal{T} \) containing the vector \( x + y \), where \( {U}_{i} \in {\mathcal{T}}_{{\gamma }_{i}} \) , \( i = 1,\ldots, n \) (Ex. 3M). Then for each index \( i \)... | Yes |
Proposition 11.26. Let \( \mathcal{E} \) be a topological vector space with linear topology \( \mathcal{T} \) and let \( \sigma \) be a pseudonorm on \( \mathcal{E} \) . The following conditions are equivalent :\n\n(i) The topology \( \mathcal{T} \) refines the topology induced by \( \sigma \) ,\n\n(ii) For an arbitrar... | Proof. Since the ball \( \{ x \in \mathcal{E} : \sigma \left( x\right) < \varepsilon \} \) is a neighborhood of 0 in the topology induced by \( \sigma \), it is clear that (i) implies (ii). To show that (ii) implies (iii) we observe that if \( x \) and \( {x}_{0} \) are vectors in \( \mathcal{E} \), then \( \left| {\si... | Yes |
Proposition 11.27. If \( {\left\{ {\sigma }_{\gamma }\right\} }_{\gamma \in \Gamma } \) is an indexed family of pseudonorms on a linear space \( \mathcal{E} \), then the collection of all sets of the form\n\n\[ \nU\left( {{\gamma }_{1},\ldots ,{\gamma }_{n};\varepsilon }\right) = \left\{ {x \in \mathcal{E} : {\sigma }_... | Proof. According to Example 3M, the collection of all finite intersections of the form\n\n\[ \nV = \left\{ {x \in \mathcal{E} : {\sigma }_{{\gamma }_{1}}\left( {x - {x}_{1}}\right) < {\varepsilon }_{1}}\right\} \cap \cdots \cap \left\{ {x \in \mathcal{E} : {\sigma }_{{\gamma }_{n}}\left( {x - {x}_{n}}\right) < {\vareps... | Yes |
Proposition 11.28. Let \( \mathcal{E} \) be a vector space, and let \( \left\{ {\sigma }_{\gamma }\right\} \) be an indexed family of pseudonorms on \( \mathcal{E} \) . Then the intersection\n\n\[ \mathcal{N} = \mathop{\bigcap }\limits_{\gamma }\left\{ {x \in \mathcal{E} : {\sigma }_{\gamma }\left( x\right) = 0}\right\... | Proof. It is clear that \( \mathcal{N} \) is a linear manifold in \( \mathcal{E} \) and that \( \mathcal{N} \) is contained in the zero space \( \mathcal{Z} = {\left( 0\right) }^{ - } \) (see Proposition 11.17 and Corollary 11.23). If \( {x}_{0} \notin \mathcal{N} \), and if, say, \( {\sigma }_{\gamma }\left( {x}_{0}\r... | Yes |
Proposition 11.30. The collection of all pseudonorms on a given linear space \( \mathcal{E} \) is a directed set in the usual ordering of real-valued functions. If \( {\sigma }_{1},\ldots ,{\sigma }_{n} \) are pseudonorms on \( \mathcal{E} \), then both \( {\sigma }_{1} + \cdots + {\sigma }_{n} \) and \( {\sigma }_{1} ... | Proof. As noted above, it is trivial to verify that \( {\sigma }_{1} + \cdots + {\sigma }_{n} \) and \( {\sigma }_{1} \vee \cdots \vee {\sigma }_{n} \) are pseudonorms, and it is visible that they both dominate the given pseudonorms. Hence, by the foregoing proposition, both \( {\sigma }_{1} + \cdots + {\sigma }_{n} \)... | Yes |
Proposition 12.1. The following conditions are equivalent for any linear transformation \( T \) of one normed space \( \mathcal{E} \) into another normed space \( \mathcal{F} \) :\n\n(i) \( T \) is continuous,\n\n(ii) \( T \) is continuous at \( x = 0 \) ,\n\n(iii) \( T \) is bounded. | Proof. Since it is clear that (i) implies (ii), it will suffice to show that (ii) implies (iii) and that (iii) implies (i). To see that (ii) implies (iii), choose \( \delta > 0 \) such that \( \parallel x\parallel = \parallel x - 0\parallel \leq \delta \) implies \( \parallel {Tx}\parallel = \parallel {Tx} - {T0}\paral... | Yes |
Proposition 12.2. If \( \mathcal{E} \) is a normed space and \( \mathcal{F} \) is a Banach space, then \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}}\right) \) is a Banach space. | Proof. Let \( \left\{ {T}_{n}\right\} \) be a Cauchy sequence in \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}}\right) \) . The inequality \( \begin{Vmatrix}{{T}_{n}x - {T}_{m}x}\end{Vmatrix} \leq \begin{Vmatrix}{{T}_{n} - {T}_{m}}\end{Vmatrix}\parallel x\parallel \) shows that for each \( x \) in \( \mathcal{E} \) the... | Yes |
Theorem 12.3. Let \( \mathcal{E},\mathcal{F} \), and \( \mathcal{G} \) be normed spaces, let \( T \) be a bounded linear transformation of \( \mathcal{E} \) into \( \mathcal{F} \), and let \( S \) be a bounded linear transformation of \( \mathcal{F} \) into \( \mathcal{G} \) . Then the product ST is a bounded linear tr... | Proof. We have \( \parallel {STx}\parallel \leq \parallel S\parallel \parallel {Tx}\parallel \leq \parallel S\parallel \parallel T\parallel \parallel x\parallel \) for every vector \( x \) in \( \mathcal{E} \) . | Yes |
Proposition 12.5. Let \( \mathcal{E} \) and \( \mathcal{F} \) be normed spaces and let \( T \) be an element of \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}}\right) \). If \( T \) is left invertible, then \( T \) is bounded below. If \( T \) is right invertible, then \( \mathcal{R}\left( T\right) = \mathcal{F} \). Mor... | Proof. If \( {RT} = {1}_{\mathcal{E}} \) then for each vector \( x \) in \( \mathcal{E} \) we have \( x = {RTx} \), and therefore\n\n\[ \parallel x\parallel \leq \parallel R\parallel \parallel {Tx}\parallel \]\n\nThus \( T \) is bounded below. (If \( \parallel R\parallel = 0 \) then \( \mathcal{E} = \left( 0\right) \),... | Yes |
Proposition 12.6. If E and F are Banach spaces, and if \( T \) is an element of \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}}\right) \) that is bounded below, then the range \( \mathcal{R} = \mathcal{R}\left( T\right) \) is closed in \( \mathcal{F} \) . Hence \( T \) is invertible if and only if \( \mathcal{R} \) is d... | Proof. Let \( T \) be bounded below by \( M, M > 0 \) . It suffices to show that \( \mathcal{R} \) is closed. Let \( \left\{ {x}_{n}\right\} \) be a sequence in \( \mathcal{E} \) such that the sequence \( \left\{ {T{x}_{n}}\right\} \) is convergent in \( \mathcal{F} \) . Since \( M\begin{Vmatrix}{{x}_{m} - {x}_{n}}\end... | Yes |
Proposition 12.7. If \( \mathcal{E} \) and \( \mathcal{F} \) are equivalent normed spaces, and if either \( \mathcal{E} \) or \( \mathcal{F} \) is a Banach space, then both must be. | Proof. Suppose without loss of generality that \( \mathcal{E} \) is complete, and that \( T : \mathcal{E} \rightarrow \mathcal{F} \) is an equivalence. Let \( \left\{ {y}_{n}\right\} \) be a Cauchy sequence in \( \mathcal{F} \), and let \( {x}_{n} = {T}^{-1}{y}_{n}, n \in \mathbb{N} \) . Then \( \begin{Vmatrix}{{x}_{m}... | Yes |
Lemma 12.8. If \( \mathcal{E} \) is a Banach space and \( T \) is a bounded operator on \( \mathcal{E} \) such that \( \parallel T\parallel < 1 \), then \( 1 - T \) is invertible and \( {\left( 1 - T\right) }^{-1} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{T}^{n} \) . | Proof. If \( \parallel T\parallel = r < 1 \), then \( \begin{Vmatrix}{T}^{n}\end{Vmatrix} \leq {r}^{n} \) for every positive integer \( n \), so the series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{T}^{n} \) is absolutely convergent in \( \mathcal{L}\left( \mathcal{E}\right) \) (see Problem 11G). If we set \[ S = \m... | Yes |
Proposition 12.9. Let & and F be Banach spaces, let T be a left [right] invertible element of \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}}\right) \), let \( R \) be a left [right] inverse of \( T \), and let \( d = \parallel R\parallel \) . Then every linear transformation \( S \) in \( \mathcal{L}\left( {\mathcal{E}... | Proof. To verify the first part of the proposition it suffices to show that if \( R \) is a left inverse of \( T \) with \( \parallel R\parallel = d \), and if \( \parallel S - T\parallel < 1/d \), then \( S \) is left invertible. But in these circumstances we have\n\n\[ \begin{Vmatrix}{{1}_{\mathcal{E}} - {RS}}\end{Vm... | Yes |
Proposition 12.10. Let \( \mathcal{A} \) be a unital Banach algebra, and let \( x \) be an element of \( \mathcal{A} \) . Then for every pair \( \alpha ,\beta \) of complex numbers in the resolvent set of \( x \) , the resolvent \( {R}_{x} \) satisfies the equation\n\n\[ \n{R}_{x}\left( \alpha \right) - {R}_{x}\left( \... | Proof. For arbitrary complex numbers \( \alpha \) and \( \beta \) we have\n\n\[ \n\left( {\alpha - x}\right) - \left( {\beta - x}\right) = \alpha - \beta . \n\]\n\nIf both \( \alpha \) and \( \beta \) belong to the resolvent set of \( x \), then, multiplying by \( {R}_{x}\left( \alpha \right) {R}_{x}\left( \beta \right... | No |
If \( \mathcal{A} \) is a unital Banach algebra and \( x \) is an arbitrary element of \( \mathcal{A} \), then the series\n\n\[ \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{x}^{n}}{{\lambda }^{n + 1}} \]\n\nconverges to \( {\left( \lambda - x\right) }^{-1} \) for every complex number \( \lambda \) such that \( \left... | Proof. If \( \left| \lambda \right| > \parallel x\parallel \) and if we define \( r = \parallel x/\lambda \parallel \), then \( r < 1 \) and \( \begin{Vmatrix}{{x}^{n}/{\lambda }^{n}}\end{Vmatrix} \leq {r}^{n} \) for every positive integer \( n \) . Hence the series (8) is absolutely convergent and therefore convergent... | Yes |
Proposition 12.12. Let \( \mathcal{A} \) be a unital Banach algebra, let \( x \) be an element of \( \mathcal{A} \), let \( {\lambda }_{0} \) be a complex number in the resolvent set of \( x \), and let \( d = \) \( \begin{Vmatrix}{{R}_{x}\left( {\lambda }_{0}\right) }\end{Vmatrix} \) . Then every scalar \( \lambda \) ... | Proof. If \( \left| {\lambda - {\lambda }_{0}}\right| < 1/d \), then\n\n\[ \n\begin{Vmatrix}{\left( {{\lambda }_{0} - \lambda }\right) {R}_{x}\left( {\lambda }_{0}\right) }\end{Vmatrix} = r < 1\n\]\n\nand\n\n\[ \n\begin{Vmatrix}{{\left( {\lambda }_{0} - \lambda \right) }^{n}{R}_{x}{\left( {\lambda }_{0}\right) }^{n}}\e... | Yes |
For an arbitrary element \( x \) of a unital Banach algebra \( \mathcal{A} \) , the spectrum \( {\sigma }_{\mathcal{A}}\left( x\right) \) is a compact set in \( \mathbb{C} \) . | Proposition 12.12 shows that the complement of \( {\sigma }_{\mathcal{A}}\left( x\right) \) is open, and we have just seen that \( {\sigma }_{\mathcal{A}}\left( x\right) \) is bounded (Prop. 12.11). | Yes |
Proposition 12.14. Let \( \mathcal{A} \) be a unital Banach algebra, let \( {x}_{0} \) be an element of \( \mathcal{A} \) , and let \( \varepsilon \) be a positive number. Then there exists a positive number \( \delta \) such that if \( x \in \mathcal{A} \) and \( \begin{Vmatrix}{x - {x}_{0}}\end{Vmatrix} < \delta \), ... | Proof. Let \( F \) denote the closed set of all those complex numbers \( \lambda \) such that \( d\left( {\lambda ,{\sigma }_{\mathcal{A}}\left( {x}_{0}\right) }\right) \geq \varepsilon \), and let \( K \) denote the compact set consisting of all those complex numbers \( \lambda \) in \( F \) such that \( \left| \lambd... | Yes |
Theorem 12.15 (Uniform Boundedness Theorem). If \( \mathcal{T} \) is a collection of bounded linear transformations of a Banach space \( \mathcal{E} \) into a normed space \( \mathcal{F} \), and if for every vector \( x \) in \( \mathcal{E} \) the set \( \mathcal{T}x = \{ {Tx} : T \in \mathcal{T}\} \) is a bounded set ... | Proof. For each positive integer \( n \) let \( {S}_{n} \) denote the set\n\n\[ \n{S}_{n} = \{ x \in \mathcal{E} : \parallel {Tx}\parallel \leq n, T \in \mathcal{T}\} .\n\]\n\nSince each \( T \) in \( \mathcal{T} \) is continuous, it is clear that \( {S}_{n} \) is a closed set. Moreover, it follows from the hypothesis ... | Yes |
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