Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Theorem 9.4.5. Let \( G \) be a Fuchsian group and let \( D\left( w\right) \) be the Dirichlet polygon with centre \( w \) . Then for almost all choices of \( w \): (1) every elliptic cycle on \( \partial D\left( w\right) \) has length 1 ; (2) every accidental cycle on \( \partial D\left( w\right) \) has length 3 ; (3)... | Proof. The proof of each part follows the same pattern: if the condition \( \left( k\right) \) fails, then \( w \) must lie in some exceptional set \( {E}_{k} \) with area zero. If \( w \) lies outside the set \( \bigcup {E}_{j} \) of zero area, then all five conditions are satisfied. We write \( D \) for \( D\left( w\... | Yes |
Lemma 9.4.6. Let \( R\left( z\right) \) be any non-constant rational function of \( z \) . Then\n\n\[ E = \{ z : R\left( z\right) \text{ is real }\} \]\n\nhas zero area. | Proof of Lemma 9.4.6. At every point of the extended plane apart from a finite set \( {z}_{1},\ldots ,{z}_{n} \), the function \( R \) is locally a homeomorphism satisfying some Lipschitz condition. Thus each \( z\left( { \neq {z}_{j}}\right) \) has a neighbourhood \( N \) with \( E \cap N \) having area zero and a cou... | Yes |
Theorem 9.6.1. Let \( G \) be a Fuchsian group acting in \( \Delta \) and let \( H \) be a subgroup of \( G \) with coset decomposition (9.6.1). Suppose that \( \Pi \) is a convex fundamental polygon for \( H \) and that a convex open polygon \( \sum \) of \( \Delta \) satisfies\n\n(1) \( \sum \) is stable under the ac... | Proof. First, \( \Pi \cap \sum \) is open and convex and its boundary has zero area. It is necessary to show that\n\n\[ \mathop{\bigcup }\limits_{{g \in G}}g\left( {\widetilde{\Pi } \cap \widetilde{\sum }}\right) = \Delta \]\n\n(9.6.3)\n\nand, if \( f \) and \( g \) are distinct elements of \( G \), then\n\n\[ g\left( ... | Yes |
Proposition 9.8.2. The sets \( {f}^{ * }\left( V\right) \) are a base for the topology of \( {X}^{ * } \) . | Proof. We know that the sets \( {f}^{ * }\left( V\right) \) are open. Suppose that \( A \) is an open subset of \( {X}^{ * } \) and that \( \langle f, x\rangle \in A \) . Writing \( \langle I, x\rangle \) as in (A6) we find that\n\n\[ \langle f, x\rangle = \left\{ {\left( {f{g}_{1},{x}_{1}}\right) ,\ldots ,\left( {f{g}... | Yes |
For \( n = 0,1,2,\ldots \), let \( {C}_{n} \) be the geodesic with end-points \( 1 + {4n} \) and \( 3 + {4n} \) and let \( {C}_{n}^{\prime } \) be its reflection in the imaginary axis. For each \( n \), let \( {g}_{n} \) be the hyperbolic element that preserves \( {H}^{2} \) and that maps the exterior of \( {C}_{n} \) ... | Now let \( D \) be the region in the second quadrant exterior to all of the \( {C}_{n}^{\prime } \) and let\n\n\[ \n{D}_{n} = \{ x + {iy} : x > 0, y > 0,{4n} < \left| z\right| < 4\left( {n + 1}\right) ,\left| {z - \left( {{4n} + 2}\right) }\right| \geq 1\} :\n\]\n\nsee Figure 10.1.1. It is clear that\n\n\[ \nP = D \cup... | Yes |
Theorem 10.2.3. A point of approximation of a Fuchsian group \( G \) cannot lie on the boundary of any convex fundamental polygon for \( G \) . | Proof. Suppose that a point of approximation \( \zeta \) lies on the boundary of a convex fundamental polygon \( P \) . By convexity, we can construct a geodesic half-ray \( L \) lying in \( P \) and ending at \( \zeta \) . By Theorem 10.2.1(3), the images \( {\left( {g}_{n}\right) }^{-1}\left( P\right) \) meet a compa... | Yes |
Theorem 10.3.1. Within the group of all isometries of the hyperbolic plane, two non-trivial conformal isometries are conjugate if and only if they have the same value of \( {\operatorname{trace}}^{2} \) . Within the group of conformal isometries, the value \( {\operatorname{trace}}^{2} \) determines two parabolic or el... | Proof. We shall prove the result in detail for the parabolic case only. Using the model \( {\mathrm{H}}^{2} \), any two parabolic isometries are conjugate (in the group of conformal isometries) to, say, \( z \mapsto z + p \) and \( z \mapsto z + q \) where \( p \) and \( q \) are real and non-zero. These are conjugate ... | No |
Theorem 10.3.2. Let \( G \) be a Fuchsian group and let \( {v}_{1},{v}_{2},\ldots \) be the parabolic and elliptic fixed points on the boundary of some convex fundamental polygon for \( G \) . Suppose that \( {g}_{j} \) generates the stabilizer of \( {v}_{j} \) : then any elliptic or parabolic element of \( G \) is con... | Proof. If \( g \) is elliptic or parabolic with fixed point \( v \), then some \( h \) in \( G \) maps \( v \) to some point on \( \partial P \) . Thus for some \( j \), we have \( h\left( v\right) = {v}_{j} \) and then \( {hg}{h}^{-1} \in \left\langle {g}_{j}\right\rangle \) . | Yes |
Corollary 10.3.3. If \( G \) is finitely generated, then \( G \) has a finite number of maximal cyclic subgroups \( \left\langle {g}_{1}\right\rangle ,\ldots ,\left\langle {g}_{n}\right\rangle \) such that any elliptic or parabolic element in \( G \) is conjugate to exactly one element in exactly one of these subgroups... | We need only observe that if \( g \) is elliptic or parabolic and if two powers of \( g \) are conjugate, say if\n\n\[ h{g}^{n}{h}^{-1} = {g}^{m} \]\n\nthen \( h \) has the same fixed points as \( g \) and so is itself a power of \( g \) : thus \( n = m \) . Note that if \( g \) is parabolic and fixes \( v \), then \( ... | No |
Theorem 10.3.4. Let \( G \) be a Fuchsian group and \( {G}_{1} \) a subgroup of index \( k \) in G. Suppose that \( G \) and \( {G}_{1} \) have \( t \) and \( {t}_{1} \) respectively, conjugacy classes of maximal parabolic cyclic subgroups. Then \( {t}_{1} \leq {kt} \) . The same result holds for elliptic elements. | Proof. Let \( D \) be a Dirichlet polygon for \( G \) in which parabolic and elliptic fixed points on \( \partial D \) have cycle length one. Thus exactly \( t \) parabolic fixed points lie in \( \partial D \) . Now express \( G \) as a coset decomposition, say\n\n\[ G = {g}_{1}{G}_{1} \cup \cdots \cup {g}_{k}{G}_{1} \... | Yes |
Theorem 10.3.5. Any non-elementary Fuchsian group contains infinitely many conjugacy classes of maximal hyperbolic cyclic subgroups. | Proof. Suppose not, then there are hyperbolic elements \( {h}_{1},\ldots ,{h}_{t} \) in \( G \) such that each hyperbolic element in \( G \) is conjugate to some power of some \( {h}_{j} \) . Let \( u \) and \( v \) be distinct limit points of \( G \) . By Theorem 5.3.8, there are hyperbolic elements \( {f}_{1},{f}_{2}... | Yes |
Theorem 10.3.6. If \( G \) is finitely generated then \( {T}_{n} \rightarrow + \infty \) as \( n \rightarrow + \infty \) . | Proof. Theorem 10.2.5 and its proof shows that every hyperbolic fixed point of \( G \) is a point of approximation and moreover, that there exists a compact subset \( K \) of \( \Delta \) such that every hyperbolic axis has an image which meets \( K \) . This means that every hyperbolic conjugacy class \( {C}_{n} \) co... | Yes |
A finitely generated Fuchsian group has only a finite number of conjugacy classes of maximal boundary hyperbolic cyclic subgroups. | A finitely generated group \( G \) has a convex fundamental polygon \( P \) with only a finite number of free sides, say \( {s}_{1},\ldots ,{s}_{n} \) . Each free side \( {s}_{j} \) lies in an interval of discontinuity \( {\sigma }_{j} \) whose stabilizer is generated by a boundary hyperbolic element, say \( {h}_{j} \)... | Yes |
Theorem 10.4.2. There is a non-elementary finitely generated Fuchsian group with signature (10.4.1) and \( {m}_{j} \geq 2 \) if and only if\n\n\[ \n{2g} - 2 + s + t + \mathop{\sum }\limits_{{j = 1}}^{r}\left( {1 - \frac{1}{{m}_{j}}}\right) > 0.\n\]\n\n(10.4.2) | The proof that (10.4.2) is a necessary condition for the existence of a group with signature (10.4.1) is a consequence of the following result.\n\nTheorem 10.4.3. Let | No |
Corollary 10.4.4. Let \( G \) be a finitely generated Fuchsian group of the first kind with signature \( \left( {g : {m}_{1},\ldots ,{m}_{r};s;0}\right) \) . Then for any convex fundamental polygon \( P \) of \( G \) , \[ \text{h-area}\left( P\right) = {2\pi }\left\lbrack {{2g} - 2 + s + \mathop{\sum }\limits_{{j = 1}}... | Proof of Theorem 10.4.3. We take \( D \) to be the Dirichlet polygon for \( G \) with centre \( w \) so \[ \mathrm{h} - \operatorname{area}\left( {D \cap N}\right) = \mathrm{h} - \operatorname{area}\left( {N/G}\right) . \] By choosing \( w \) appropriately, we may assume that each elliptic and parabolic cycle on \( \pa... | No |
Theorem 10.5.1. Let \( G \) be a finitely generated Fuchsian group of the first kind and let \( P \) be any convex fundamental polygon for \( G \) . Suppose that \( P \) has \( N \) sides (where no side is paired with itself).\n\n(i) If \( G \) has signature \( \left( {g : {m}_{1},\ldots ,{m}_{n}}\right) \) where possi... | Proof. Suppose that \( P \) has elliptic or parabolic cycles \( {C}_{1},\ldots ,{C}_{n} \) and accidental cycles \( {C}_{n + 1},\ldots ,{C}_{n + A} \) : either (but not both) of these sets of cycles may be absent. In general, we let \( \left| C\right| \) denote the number of points in the cycle \( C \) .\n\nNow\n\n\[ \... | Yes |
Example 10.6.2. Let \( {T}_{1} \) and \( {T}_{2} \) be the two triangles illustrated in Figure 10.6.1: the corresponding groups are\n\n\[ \n{G}_{1} = \left\langle {{\sigma }_{1},{\sigma }_{2},\eta }\right\rangle \n\]\n\nof type \( \left( {0,\pi /2,\pi /3}\right) \) and\n\n\[ \n{G}_{2} = \left\langle {{\sigma }_{1},{\si... | Clearly,\n\n\[ \n\eta {\sigma }_{1} = {\sigma }_{1}\tau \n\]\n\nso \( \eta \in {G}_{2} \) and \( \tau \in {G}_{1} \) ; thus \( {G}_{1} = {G}_{2} \) . In fact, the subgroup of conformal isometries of this group is the Modular group and so \( {G}_{1} \) is itself discrete. Note that\n\n\[ \n\mathrm{h} - \operatorname{are... | Yes |
Theorem 11.2.5. Let \( G \) be a non-elementary Fuchsian group without elliptic elements. If \( \left( {v}_{j}\right) \) is an accidental cycle of vertices on the boundary of the Dirichlet polygon with centre \( w \), then\n\n\[ \n\cosh \rho \left( {w,{v}_{j}}\right) \geq \sqrt{2}\text{.}\n\] | Proof of Theorem 11.2.5. The cycle \( \left( {v}_{j}\right) \) lies on a circle \( C \), say \( \{ z : \rho \left( {z, w}\right) = r\} \) and contains at least three vertices with, say,\n\n\[ \n{v}_{2} = g\left( {v}_{1}\right) ,\;{v}_{3} = h\left( {v}_{1}\right) .\n\]\n\nLet \( {G}_{0} \) be the group generated by \( g... | Yes |
Given any integer \( k \) with \( k \geq 2 \) we can construct a Fuchsian group \( G \) acting on \( \Delta \) which has as its fundamental domain a regular polygon with \( {4k} \) sides and all vertices lying in one accidental cycle (see Section 10.4). | Referring to the proof of Theorem 11.2.4(i), we find that \( {\alpha }_{j} = {\theta }_{j} = \pi /{4k} \) so in this case, equality holds in (i). Thus (at least for \( n \) of the form \( {4k} \) ), Theorem 11.2.4(i) is best possible. | No |
Theorem 11.2.7. Let \( D \) be a fundamental polygon for a Fuchsian group \( G \) and suppose that \( D \) contains two points \( {w}_{1} \) and \( {w}_{2} \) on the circle at infinity. Let \( L \) be the geodesic joining \( {w}_{1} \) and \( {w}_{2} \) . If \( v \) is an elliptic fixed point of \( G \) of order \( n \... | Proof. The triangle with vertices \( {w}_{1},{w}_{2} \) and \( v \) lies in \( D \) and so the interior angle of this triangle at \( v \) cannot exceed \( {2\pi }/n \) . This means that \( v \) cannot be too close to \( L \) : the numerical details are left to the reader. | No |
Proposition 11.3.2. Let \( G \) be a Fuchsian group with parabolic elements. If \( G \) has a fundamental domain with h-area less than \( \pi \), then \( G \) has one of the signatures \( \left( {0 : 2, q,\infty }\right) \) where \( 3 \leq q \leq + \infty \) or \( \left( {0 : 3, q,\infty }\right) \) where \( q = 3,4 \)... | Proof. As the fundamental domain has finite area, \( G \) has signature \( \left( {k : {m}_{1}}\right. \) , \( \ldots ,{m}_{n},\infty ) \) say, the \( \infty \) being present as \( G \) is known to include parabolic elements. From Section 10.4 we deduce that\n\n\[ \n{2\pi }\left\lbrack {{2k} - 2 + \mathop{\sum }\limits... | Yes |
Theorem 11.3.3. Let \( {G}_{0} \) be a Hecke group and let \( G \) be a Fuchsian group containing \( {G}_{0} \) . Then \( G = {G}_{0} \) . | Proof. We may suppose that \( G \) acts on \( {H}^{2} \) so\n\n\[ k\mathrm{\;h}\text{-area}\left( {{H}^{2}/G}\right) = \mathrm{h}\text{-area}\left( {{H}^{2}/{G}_{0}}\right) \text{,}\]\n\n(11.3.2)\n\nwhere \( {G}_{0} \) is of index \( k \) in \( G \) . By assumption, \( {G}_{0} \) has signature \( \left( {0 : 2, p,\inft... | Yes |
Theorem 11.4.3. Let \( g \) be a rotation of angle \( {2\pi }/n\left( {n \geq 3}\right) \) about some point in the hyperbolic plane and suppose that \( f \) and \( g \) generate a non-elementary Fuchsian group. Then apart from certain Triangle groups (which are listed in the proof), (1) \( \operatorname{trace}\left\lbr... | Proof. We may suppose that \( f \) and \( g \) act on \( \Delta \) and that in terms of matrices, \[ g = \left( \begin{matrix} {e}^{{i\pi }/n} & 0 \\ 0 & {e}^{-{i\pi }/n} \end{matrix}\right) ,\;f = \left( \begin{array}{ll} a & \bar{c} \\ c & \bar{a} \end{array}\right) , \] where \( {\left| a\right| }^{2} - {\left| c\ri... | Yes |
Theorem 11.5.1. The absolute value of the trace of any of the isometries\n\n\\[ \n{fgh},{hfg},{ghf},{hgf},{fhg},{gfh} \n\\]\n\nis equal to \\( {2\\lambda } \\) . | Proof. First, \\( \\left| {\\operatorname{trace}\\left( {fgh}\\right) }\\right| \\) is invariant under cyclic permutations of \\( f, g \\) and \\( h \\) : for example,\n\n\\[ \n\\left| {\\operatorname{trace}\\left( {fgh}\\right) }\\right| = \\left| {\\operatorname{trace}h\\left( {fgh}\\right) {h}^{-1}}\\right| \n\\]\n\... | Yes |
Theorem 11.5.2. Let \( f, g \) and \( h \) be elliptic elements of order two which generate a non-elementary group \( G \) and let \( \lambda \) be given by (11.5.1).\n\n(1) If \( \lambda > 1 \) then \( G \) is discrete and has signature \( \left( {0 : 2,2,2;0;1}\right) \).\n\n(2) If \( \lambda = 1 \) then \( G \) is d... | A construction of a fundamental domain for each discrete \( G \) will arise in the proof and it will be apparent that every value of \( \lambda \) given in Theorem 11.5.2 does give rise to a discrete group. | Yes |
Theorem 11.6.1. Let \( g \) and \( h \) be isometries and suppose that \( \langle g, h\rangle \) is discrete and non-elementary.\n\n(1) If \( g \) and \( h \) are parabolic, then \( P\left( {g, h}\right) \geq \frac{1}{4} \) . If, in addition, \( \langle g, h\rangle \) is not a Triangle group, then \( P\left( {g, h}\rig... | Proof. Let \( g \) be parabolic and let \( h \) be parabolic or hyperbolic. We may suppose that \( g \) and \( h \) act on \( {H}^{2} \) and that\n\n\[ g\left( z\right) = z + 1,\;h\left( z\right) = \frac{{az} + b}{{cz} + d},\;{ad} - {bc} = 1. \]\n\nAs \( \langle g, h\rangle \) is non-elementary, \( c \neq 0 \) . Now \(... | Yes |
The isometries \( g, h \) and \( f \) given by\n\n\[ g\left( z\right) = z + 1,\;h\left( z\right) = \frac{z}{z + 1},\;f\left( z\right) = \frac{{2z} + 3}{z + 2} \]\n\nare parabolic, parabolic and hyperbolic respectively and generate a discrete group (a subgroup of the Modular group). | A computation using (11.6.1) with \( z = {iy} \) gives\n\n\[ \sinh \frac{1}{2}\rho \left( {z,{gz}}\right) \sinh \frac{1}{2}\rho \left( {z,{hz}}\right) = \frac{1}{4} \]\n\nand\n\n\[ \sinh \frac{1}{2}\rho \left( {z,{gz}}\right) \sinh \frac{1}{2}\rho \left( {z,{fz}}\right) = \frac{1}{4} + \left( {3/4{y}^{2}}\right) \]\n\n... | Yes |
Let \( g\left( z\right) = z + 1 \) and let \( h \) be the reflection in \( \left| {z + t}\right| = t \) followed by reflection in \( x = 0 \) where \( 0 < t < \frac{1}{4} \) . Thus \( h \) is parabolic and fixes the origin: in fact,\n\n\[ h\left( z\right) = \frac{z}{\left( {z/t}\right) + 1}. \] | Using (11.6.1), we see that when \( z = {iy} \) ,\n\n\[ \sinh \frac{1}{2}\rho \left( {z,{gz}}\right) \sinh \frac{1}{2}\rho \left( {z,{hz}}\right) = 1/{4t}. \] | No |
Let \( g\left( z\right) = z + 1 \) and let \( h \) be an elliptic element of order two fixing the point \( {iv} \) where \( 0 < v < \frac{1}{2} \). Then \( \langle g, h\rangle \) is discrete and nonelementary: for example, | \[ \left\{ {z \in {H}^{2} : \left| {\operatorname{Re}\left\lbrack z\right\rbrack }\right| < \frac{1}{2},\left| z\right| > v}\right\} \] is a fundamental domain for \( \langle g, h\rangle \). Now write \( f = {gh} \): then \( f \) is hyperbolic and is a reflection in \( \left| z\right| = v \) followed by the reflection ... | Yes |
Theorem 11.6.6. Let \( g \) and \( h \) be elliptic elements of orders \( p \) and \( q \) respectively and suppose that \( \langle g, h\rangle \) is discrete and non-elementary. Then\n\n\[ M\left( {g, h}\right) \geq {\left\lbrack \frac{4{\cos }^{2}\left( {\pi /7}\right) - 3}{8\cos \left( {\pi /7}\right) + 7}\right\rbr... | If, in addition, \( \langle g, h\rangle \) is not a Triangle group, then\n\n\[ M\left( {g, h}\right) \geq {\left( \frac{{\left\lbrack \cos \left( \pi /p\right) + \cos \left( \pi /q\right) \right\rbrack }^{2}}{4 - {\left\lbrack \cos \left( \pi /p\right) - \cos \left( \pi /q\right) \right\rbrack }^{2}}\right) }^{1/2} \ge... | Yes |
Theorem 11.6.7. Let \( g \) be elliptic of order \( p \) with fixed point \( u \), let \( h \) be elliptic of order \( q \) with fixed point \( v \) and suppose that \( \langle g, h\rangle \) is discrete, non-elementary but not a Triangle group. Then \[ \cosh \rho \left( {u, v}\right) > \frac{1 + \cos \left( {\pi /p}\r... | Proof of THEOREM 11.6.7. Some \( {g}_{1} \) in \( \langle g\rangle \) has angle of rotation \( {2\pi }/p \), some \( {h}_{1} \) in \( \langle h\rangle \) has angle of rotation \( {2\pi }/q \) and \( \langle g, h\rangle = \left\langle {{g}_{1},{h}_{1}}\right\rangle \) . Thus we may assume that \( g \) and \( h \) have a... | Yes |
Theorem 11.6.8. Let \( g \) and \( h \) be hyperbolic elements with axes and translation lengths \( {A}_{g},{A}_{h},{T}_{g} \) and \( {T}_{h} \) respectively. Suppose that \( \langle g, h\rangle \) is discrete and nonelementary and that \( {A}_{g} \) and \( {A}_{h} \) cross at an angle \( \theta \) . Then\n\n(1)\n\n\[ ... | Proof. Let \( u \) be the point where \( {A}_{g} \) and \( {A}_{h} \) cross and construct points \( v \) and \( w \) on \( {A}_{g} \) and \( {A}_{h} \) respectively such that \( \rho \left( {u, v}\right) = \frac{1}{2}{T}_{g},\rho \left( {u, w}\right) = \frac{1}{2}{T}_{h} \) and such that the triangle with vertices \( u... | Yes |
Theorem 11.6.9. Let \( g \) and \( h \) be hyperbolic with axes and translation lengths \( {A}_{g},{A}_{h},{T}_{g} \) and \( {T}_{h} \) respectively. Suppose that \( \langle g, h\rangle \) is discrete and nonelementary and that no images of \( {A}_{g} \) and \( {A}_{h} \) cross. Then\n\n\[ \sinh \left( {\frac{1}{2}{T}_... | If \( \langle g, h\rangle \) has no elliptic elements, we can replace \( - \frac{1}{2}{by} + 1 \) (and the lower bound by 2).\n\nIf \( g \) is a simple hyperbolic element in \( \langle g, h\rangle \) this result can be applied with \( h \) being any conjugate, say \( {fg}{f}^{-1} \), of \( g \) . Thus (by elementary ma... | No |
Corollary 11.6.10. If \( g \) and \( h \) are hyperbolic elements generating a discrete non-elementary group and if \( g \) is a simple hyperbolic element in this group, then for all \( f \) in \( \langle g, h\rangle \), either \( f\left( {A}_{g}\right) = {A}_{g} \) or\n\n\[ \sinh \left( {\frac{1}{2}{T}_{g}}\right) \si... | This bound is best possible.\n\nThe next example shows that the lower bound of \( \frac{1}{2} \) is best possible. | No |
Construct the polygon \( D \) as in Figure 11.6.3 where \( f \) (elliptic of order two) and \( g \) (hyperbolic) pair the sides of \( D \) . By Poincaré’s Theorem, \( D \) is a fundamental polygon for \( \langle f, g\rangle \) and as \( g \) pairs the sides of \( D, g \) must be a simple hyperbolic element. Finally, | \[ \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \frac{1}{2}\rho \left( {{A}_{g}, f{A}_{g}}\right) = \sinh \frac{1}{2}\rho \left( {L,{L}^{\prime }}\right) \sinh \rho \left( {0,{A}_{g}}\right) = \cos \left( {\pi /3}\right) \text{.} \] | Yes |
Theorem 11.6.12. Let \( g \) and \( h \) be hyperbolic elements which generate a discrete non-elementary group. Then \( P\left( {g, h}\right) \geq \cos \left( {{3\pi }/7}\right) \) . | Proof. If the axes of \( g \) and \( h \) cross at \( w \), say, then obviously (using the notation of Theorems 11.6.8 and 11.6.9)\n\n\[ \begin{array}{l} P\left( {g, h}\right) = \sinh \frac{1}{2}\rho \left( {w,{gw}}\right) \sinh \frac{1}{2}\rho \left( {w,{hw}}\right) \end{array} \]\n\n\[ = \sinh \left( {\frac{1}{2}{T}_... | Yes |
Theorem 11.6.13. Let \( g \) be hyperbolic and let \( h \) be elliptic of order \( q\left( {q \geq 2}\right) \) . If \( \langle g, h\rangle \) is discrete and non-elementary, then \( M\left( {g, h}\right) \geq 1/\sqrt{8} \) . | Proof. If \( g \) is a non-simple hyperbolic element of \( \langle g, h\rangle \) then (from Theorem 11.6.8)\n\n\[ M\left( {g, h}\right) \geq \sinh \left( {\frac{1}{2}{T}_{g}}\right) \]\n\n\[ \geq {\left\lbrack \cos \left( 3\pi /7\right) \right\rbrack }^{1/2} \]\n\n\[ > 1/\sqrt{8}\text{.} \]\n\nWe may now assume that \... | Yes |
Theorem 11.7.1. Let \( G \) be a Fuchsian group without elliptic elements, and suppose that \( g \) and \( h \) are in \( G \) . (1) If \( g \) and \( h \) are parabolic elements with district fixed points, then \( {\sum }_{g} \) and \( {\sum }_{h} \) are disjoint. | Proof. For a Fuchsian group without elliptic elements, we have (Theorem 8.3.1) \[ \sinh \frac{1}{2}\rho \left( {z,{gz}}\right) \sinh \frac{1}{2}\rho \left( {z,{hz}}\right) \geq 1, \] whenever \( \langle g, h\rangle \) is non-elementary. In view of (11.7.1), this proves (1). For a geometric proof of (1), we may assume t... | Yes |
Theorem 11.7.2. Let \( G \) be a non-elementary, non-Triangle Fuchsian group. If \( g \) and \( h \) are elliptic or parabolic elements in \( G \), then either \( \langle g, h\rangle \) is cyclic or the canonical regions \( {\sum }_{g} \) and \( {\sum }_{h} \) are disjoint. | Proof. We may assume that \( g \) and \( h \) are primitive (this can only increase the size of \( {\sum }_{g} \) and \( {\sum }_{h} \) ). Construct the geodesic \( L \) through (or ending at) the fixed point \( u \) of \( g \) and the fixed point \( v \) of \( h \) . Construct geodesics \( {L}_{1} \) and \( {L}_{2} \)... | Yes |
Theorem 1. If \( X \) and \( Y \) are finite-dimensional normed linear spaces of the same dimension, then they are isomorphic. | Proof. We show that if \( X \) has dimension \( n \), the \( X \) is isomorphic to \( {l}_{1}^{n} \) . Recall that the norm of an \( n \) -tuple \( \left( {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right) \) in \( {l}_{1}^{n} \) is given by\n\n\[ \begin{Vmatrix}\left( {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right) \end{Vmatrix} = \le... | Yes |
Corollary 2. Finite-dimensional normed linear spaces are complete. | In fact, a normed linear space isomorphism is Lipschitz continuous in each direction and so must preserve completeness; by Theorem 1 all \( n \) -dimensional spaces are isomorphic to the Banach space \( {l}_{1}^{n} \) . | Yes |
Lemma. Let \( Y \) be a proper closed linear subspace of the normed linear space \( X \) and \( 0 < \theta < 1 \) . Then there is an \( {x}_{\theta } \in {S}_{X} \) for which \( \begin{Vmatrix}{{x}_{\theta } - y}\end{Vmatrix} > \theta \) for every \( y \in Y \) . | Proof. Pick any \( x \in X \smallsetminus Y \) . Since \( Y \) is closed, the distance from \( x \) to \( Y \) is positive, i.e., \[ 0 < d = \inf \{ \parallel x - z\parallel : z \in Y\} < \frac{d}{\theta }; \] therefore, there is a \( z \in Y \) such that \[ \parallel x - z\parallel < \frac{d}{\theta }. \] Let \[ {x}_{... | Yes |
In order for each closed bounded subset of the normed linear space \( X \) to be compact, it is necessary and sufficient that \( X \) be finite dimensional. | Should the dimension of \( X \) be \( n \), then \( X \) is isomorphic to \( {l}_{2}^{n} \) (Theorem 1); therefore, the compactness of closed bounded subsets of \( X \) follows from the classical Heine-Borel theorem.\n\nShould \( X \) be infinite dimensional, then \( {S}_{X} \) is not compact, though it is closed and b... | Yes |
Theorem 1. If \( K \) is a convex subset of the normed linear space \( X \), then the closure of \( K \) in the norm topology coincides with the weak closure of \( K \) . | Proof. There are no more open sets in the weak topology than there are in the norm topology; consequently, the norm closure is harder to get into than the weak closure. In other words \( {\bar{A}}^{\parallel \cdot \parallel } \subseteq {\bar{A}}^{\text{weak }} \) . If \( K \) is a convex set and if there were a point \... | Yes |
Lemma 2. Let \( F \) be a finite-dimensional linear subspace of the infinite-dimensional Banach space \( X \), and let \( \varepsilon > 0 \) . Then there is an \( x \in X \) such that \( \parallel x\parallel = 1 \) and\n\n\[ \parallel y\parallel \leq \left( {1 + \varepsilon }\right) \parallel y + {\lambda x}\parallel \... | Proof. Assuming (as we may) that \( \varepsilon < 1 \), pick a finite \( \varepsilon /2 \) net \( \left\{ {{y}_{1},\ldots ,{y}_{k}}\right\} \) for \( {S}_{F} \) and select \( {y}_{1}^{ * },\ldots ,{y}_{k}^{ * } \) in \( {S}_{{X}^{ * }} \) so that \( {y}_{i}^{ * }{y}_{i} = 1 \) for \( i = 1,2,\ldots, k \) . Take any \( ... | Yes |
Corollary 3. Every infinite-dimensional Banach space contains an infinite-dimensional closed linear subspace with a basis. | Proof. Let \( X \) be the ambient space and \( \varepsilon > 0 \) . Choose a sequence \( \left( {\varepsilon }_{n}\right) \) of positive numbers such that \( \mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 + {\varepsilon }_{n}}\right) \leq 1 + \varepsilon \) . Take \( {x}_{1} \in {S}_{X} \) and pick \( {x}_{2} \in ... | Yes |
Theorem 5. The bases \( \left( {x}_{n}\right) \) and \( \left( {y}_{n}\right) \) are equivalent if and only if there is an isomorphism between \( X \) and \( Y \) that carries each \( {x}_{n} \) to \( {y}_{n} \) . | Proof. Recall that in our earlier comments about bases-we renormed \( \cdot X \) by taking any \( x = \mathop{\sum }\limits_{n}{s}_{n}{x}_{n} \) and defining \( \parallel \mid x\parallel \mid \) by\n\n\[ \parallel \left| x\right| \parallel \mathrel{\text{:=}} \mathop{\sup }\limits_{n}\begin{Vmatrix}{-\mathop{\sum }\lim... | No |
Theorem 6. The following statements regarding a formal series \( \mathop{\sum }\limits_{n}{x}_{n} \) in a Banach space are equivalent:\n\n1. \( \mathop{\sum }\limits_{n}{x}_{n} \) is wuC.\n\n2. There is a \( C > 0 \) such that for any \( \left( {t}_{n}\right) \in {l}_{\infty } \)\n\n\[ \mathop{\sup }\limits_{n}\begin{V... | Proof. Suppose 1 holds and define \( T : {X}^{ * } \rightarrow {l}_{1} \) by\n\n\[ T{x}^{ * } = \left( {{x}^{ * }{x}_{n}}\right) \]\n\n\( T \) is a well-defined linear map with a closed graph; therefore, \( T \) is bounded. From this we see that for any \( \left( {t}_{n}\right) \in {B}_{{l}_{\infty }} \) and any \( {x}... | Yes |
Corollary 7. A basic sequence for which \( \mathop{\inf }\limits_{n}\begin{Vmatrix}{x}_{n}\end{Vmatrix} > 0 \) and \( \mathop{\sum }\limits_{n}{x}_{n} \) is wuC is equivalent to the unit vector basis of \( {c}_{0} \) . | Proof. If \( \left( {x}_{n}\right) \) is a basic sequence and \( \mathop{\sum }\limits_{n}{t}_{n}{x}_{n} \) is convergent, then \( \left( {\mathop{\sum }\limits_{{k = 1}}^{n}{t}_{k}{x}_{k}}\right) \) is a Cauchy sequence. Therefore, letting \( n \) tend to infinity, the sequence\n\n\[ \left| {t}_{n}\right| \begin{Vmatr... | Yes |
Theorem 9. Let \( \left( {z}_{n}\right) \) be a basic sequence in the Banach space \( X \), and suppose \( \left( {z}_{n}^{ * }\right) \) is the sequence of coefficient functionals (extended to all of \( X \) in a Hahn-Banach fashion). Suppose \( \left( {y}_{n}\right) \) is a sequence in \( X \) for which \( \mathop{\s... | In fact, if we define \( T : X \rightarrow X \) by\n\n\[ \n{Tx} = \mathop{\sum }\limits_{n}{z}_{n}^{ * }\left( x\right) \left( {{z}_{n} - {y}_{n}}\right) \n\] \n\nthen \( \parallel T\parallel \leq \mathop{\sum }\limits_{n}\begin{Vmatrix}{z}_{n}^{ * }\end{Vmatrix}\begin{Vmatrix}{{z}_{n} - {y}_{n}}\end{Vmatrix} < 1 \) . ... | Yes |
In order that each series \( \mathop{\sum }\limits_{n}{x}_{n}^{ * } \) in the dual \( {X}^{ * } \) of a Banach space \( X \) for which \( \mathop{\sum }\limits_{n}\left| {{x}_{n}^{ * }x}\right| < \infty \) for each \( x \in X \) be unconditionally convergent, it is both necessary and sufficient that \( {X}^{ * } \) con... | Proof. If \( {X}^{ * } \) contains an isomorphic copy of \( {l}_{\infty } \), then it contains a weak* isomorphic copy \( Z \) of \( {l}_{\infty } \) as described in part 3 of Theorem 10. Looking at the unit vectors of \( {c}_{0} \) as they appear in \( Z \), they look just as they do in \( {l}_{\infty } : \mathop{\sum... | Yes |
Theorem 12. Let \( \left( {z}_{n}\right) \) be a basic sequence in the Banach space \( X \) with coefficient functionals \( \left( {z}_{n}^{ * }\right) \) . Suppose that there is a bounded linear projection \( P : X \rightarrow X \) onto the closed linear span \( \left\lbrack {z}_{n}\right\rbrack \) of the \( {z}_{n} \... | Proof. Since \( P \) is a linear projection with nontrivial range, \( \parallel P\parallel \geq 1 \) . It follows then from Theorem 9 that \( \left( {y}_{n}\right) \) is a basic sequence equivalent to \( \left( {z}_{n}\right) \) . The condition set forth in the hypotheses is easily seen to be just what is needed to pro... | Yes |
1. Let \( \Omega \) be any topological space each closed subset of which is of the second category in itself. Then any bounded scalar-valued function on \( \Omega \) which is the pointwise limit of a sequence of continuous scalar-valued functions on \( \Omega \) has a point of continuity in each nonvoid closed subset o... | The proof of part 1 of Theorem 1 is easy. We need to recall that the members of \( C{\left( \Omega \right) }^{ * } \) act on \( C\left( \Omega \right) \) like integration via regular Borel measures on \( \Omega \) . This in mind, suppose \( f,{f}_{n} \in C\left( \Omega \right) \left( {n \geq 1}\right) \) satisfy \( f\l... | Yes |
Theorem 3 (R. S. Phillips). Let \( Y \) be a linear subspace of the Banach space \( X \) and suppose \( T : Y \rightarrow {l}_{\infty } \) is a bounded linear operator. Then \( T \) may be extended to a bounded linear operator \( S : X \rightarrow {l}_{\infty } \) having the same norm as \( T \) . | Proof. A bit of thought brings one to observe that the operator \( T \) must be of the form\n\n\[ \n{Ty} = \left( {{y}_{n}^{ * }y}\right) \n\]\n\nfor some bounded sequence \( \left( {y}_{n}^{ * }\right) \) in \( {Y}^{ * } \) . If we let \( {x}_{n}^{ * } \) be a Hahn-Banach extension of \( {y}_{n}^{ * } \) to all of \( ... | Yes |
Theorem 5. If \( X \) is any Banach space and \( T : X \rightarrow {l}_{1} \) is a bounded linear operator of \( X \) onto \( {l}_{1} \) then \( X \) contains a complemented subspace that is isomorphic to \( {l}_{1} \) . Moreover, among the separable infinite-dimensional Banach spaces, the above assertion characterizes... | As is only fair, we start with the proof of the first assertion. Suppose \( T : X \rightarrow {l}_{1} \) is as advertised. By the open mapping theorem there is a bounded sequence \( \left( {x}_{n}\right) \) in \( X \) such that \( T{x}_{n} = {e}_{n} \) . Consider the bounded linear operator \( S : {l}_{1} \rightarrow X... | No |
Theorem 7. The dual of \( B\left( \sum \right) \) is identifiable with the space \( \operatorname{ba}\left( \sum \right) \) under the correspondence\n\n\[ \n{x}^{ * } \in B{\left( \sum \right) }^{ * } \leftrightarrow \mu \in \mathrm{{ba}}\left( \sum \right)\n\]\n\ngiven by\n\n\[ \n{x}^{ * }f = \int {fd\mu }\n\]\n\nFurt... | For instance, suppose \( \mu \in \operatorname{ba}\left( \sum \right) \) and let \( \left( {A}_{n}\right) \) be a sequence of disjoint members of \( \sum \) . For each \( n \) we have\n\n\[ \n\mathop{\sum }\limits_{{k = 1}}^{n}\left| {\mu \left( {A}_{k}\right) }\right| \leq \parallel \mu {\parallel }_{1}\n\]\n\nso that... | No |
Theorem 8. \( \left( {\sum ,{d}_{\lambda }}\right) \) is a complete pseudometric space on which the operations \( \left( {A, B}\right) \rightarrow A \cup B,\left( {A, B}\right) \rightarrow \left( {A \cap B}\right) \), and \( A \rightarrow {A}^{c} \) are all continuous (the first two as functions of two variables). | To see that \( \left( {\sum ,{d}_{\lambda }}\right) \) is complete, notice that for \( A, B \in \sum \) ,\n\n\[ \n{d}_{\lambda }\left( {A, B}\right) = {\begin{Vmatrix}{c}_{A} - {c}_{B}\end{Vmatrix}}_{{L}_{1}\left( \lambda \right) }.\n\]\n\nTherefore, if \( \left( {A}_{n}\right) \) is a \( {d}_{\lambda } \) -Cauchy sequ... | Yes |
Theorem 9. Let \( \mathcal{K} \) be a family of finitely additive scalar-valued measures defined on \( \sum \) . Then the following are equivalent (TFAE):\n\n1. \( \mathcal{K} \) is equi- \( \lambda \) -continuous at some \( E \in \sum \) .\n\n2. \( \mathcal{K} \) is equi- \( \lambda \) -continuous at \( \varnothing \)... | Proof. Suppose 1 holds. Let \( \varepsilon > 0 \) be given and choose \( \delta > 0 \) so that should \( B \in \sum \) be within \( \delta \) of \( E \), then \( \left| {\mu \left( B\right) - \mu \left( E\right) }\right| \leq \varepsilon \) for all \( \mu \in \mathcal{K} \) .\n\nNotice that if \( A \in \sum \) and \( \... | Yes |
Theorem 10. Let \( \mathcal{K} \subseteq \operatorname{ca}\left( \sum \right) \). Then TFAE:\n\n1. If \( \left( {E}_{n}\right) \) is a sequence of disjoint members of \( \sum \), then for each \( \varepsilon > 0 \) there is an \( {n}_{\varepsilon } \) such that for \( m \geq n \geq {n}_{\varepsilon } \), \n\n\[ \left| ... | Proof. The proof is purely formal and proceeds as with one measure at a time with the phrase \ | No |
Theorem 11. A sequence \( \left( {\mu }_{n}\right) \) in \( \mathrm{{ca}}\left( \sum \right) \) converges weakly to \( \mu \in \mathrm{{ca}}\left( \sum \right) \) if and only if for each \( E \in \sum ,\mu \left( E\right) = \mathop{\lim }\limits_{n}{\mu }_{n}\left( E\right) \) . | Proof. Since the functional \( \nu \rightarrow \nu \left( E\right) \) belongs to \( \operatorname{ca}{\left( \sum \right) }^{ * } \) for each \( E \in \sum \) , the necessity of \( \mu \left( E\right) = \mathop{\lim }\limits_{n}{\mu }_{n}\left( E\right) \) for each \( E \in \sum \) is clear.\n\nSuppose for the sake of ... | Yes |
Theorem 12. Weakly Cauchy sequences in \( \mathrm{{ca}}\left( \sum \right) \) are weakly convergent. Consequently, for any \( \lambda \in {\mathrm{{ca}}}^{ + }\left( \sum \right) \), weakly Cauchy sequences in \( {L}_{1}\left( \lambda \right) \) are weakly convergent. | Proof. Let \( \left( {\mu }_{n}\right) \) be a weakly Cauchy sequence in \( \operatorname{ca}\left( \sum \right) \) . Since each \( E \in \sum \) determines the member \( \nu \rightarrow \nu \left( E\right) \) of \( \operatorname{ca}{\left( \sum \right) }^{ * },\mathop{\lim }\limits_{n}{\mu }_{n}\left( E\right) = \mu \... | Yes |
Theorem 14 (Dieudonné-Grothendieck). Let \( \Omega \) be a compact Hausdorff space and \( \sum \) be the \( \sigma \) -field of Borel subsets of \( \Omega \). Suppose \( \mathcal{K} \) is a bounded subset of \( \operatorname{rca}\left( \sum \right) \). In order for \( \mathcal{K} \) to be relatively weakly compact, it ... | Proof. Necessity is clear from the uniform countable additivity of relatively weakly compact subsets of \( \operatorname{ca}\left( \sum \right) \). To establish sufficiency we will mount a two pronged attack by proving that should the bounded set \( \mathcal{K} \) satisfy \( \mathop{\lim }\limits_{n}\mu \left( {O}_{n}\... | Yes |
Theorem 3 (N. and V. Gurarii). If the normalized Schauder basis \( \left( {x}_{n}\right) \) spans a uniformly convex space \( X \), then there is a \( p > 1 \) and an \( A > 0 \) such that \( \mathop{\sum }\limits_{n}{a}_{n}{x}_{n} \in X \) whenever \( \left( {a}_{n}\right) \in {l}_{p} \) | \[ \begin{Vmatrix}{\mathop{\sum }\limits_{n}{a}_{n}{x}_{n}}\end{Vmatrix} \leq A{\begin{Vmatrix}\left( {a}_{n}\right) \end{Vmatrix}}_{p} \] | Yes |
Lemma 5. \( \delta \) is a nondecreasing function of \( \varepsilon \) in \( \left\lbrack {0,2}\right\rbrack \) . | Proof. Let \( 0 \leq {\varepsilon }_{1} < {\varepsilon }_{2} \leq 2 \) .\n\nPick \( x, y \in {S}_{X} \) so that \( \parallel x - y\parallel = {\varepsilon }_{2} \) and \( \parallel x + y\parallel = 2\left( {1 - \delta \left( {\varepsilon }_{2}\right) }\right) \) . Let \( c = \) \( \left( {{\varepsilon }_{2} - {\varepsi... | Yes |
Theorem 1. Suppose the closed convex hull \( K \) of a closed set \( F\left( { \subseteq E}\right) \) is compact. Then each regular Borel probability measure \( \mu \) on \( F \) has a unique barycenter in \( K \) . | Proof. The restriction \( {\left. f\right| }_{F} \) of any \( f \in {E}^{ * } \) to \( F \) is plainly \( \mu \) -integrable for any regular Borel probability measure \( \mu \) on \( F \) . Take any such \( \mu \) . We claim that the hyperplanes\n\n\[ \n\left\{ {x \in E : f\left( x\right) = {\int }_{F}{fd\mu }}\right\}... | Yes |
For any compact subset \( C \) in \( E \), a point \( x \) of \( E \) belongs to the closed convex hull of \( C \) if and only if there exists a regular Borel probability measure \( \mu \) on \( C \) whose barycenter (exists and) is \( x \) . | If \( \mu \) is a regular Borel probability measure on \( C \) and has \( x \) as its barycenter, then for any \( f \in {E}^{ * } \) we know\n\n\[ f\left( x\right) = {\int }_{C}{fd\mu } \leq \sup f\left( C\right) \leq \sup f\left( {\overline{\operatorname{co}}C}\right) .\n\]\nWere \( x \) not in \( \overline{\operatorn... | Yes |
Theorem 3 (Bauer’s Characterization of Extreme Points). Let \( K \) be a nonempty compact convex subset of \( E \) . A point \( x \) of \( K \) is an extreme point of \( K \) if and only if \( {\delta }_{x} \) is the only regular Borel probability measure on \( K \) that represents \( x \) . | Proof. If \( x \in K \) is not an extreme point, then there are \( y, z \in K \) with \( y \neq z \) so that \( x = \frac{1}{2}y + \frac{1}{2}z \) . Plainly, \( \frac{1}{2}{\delta }_{v} + \frac{1}{2}{\delta }_{z} \) is a regular Borel probability measure on \( K \) that represents \( x \) and differs from \( {\delta }_... | Yes |
Corollary 4 (Milman’s Converse to the Krein-Milman Theorem). Let \( K \) be a compact convex subset of \( E \) . If \( K \) is the closed convex hull of a set \( Z \), then every extreme point of \( K \) lies in \( Z \)’s closure. | Proof. Suppose \( x \) is an extreme point of \( K = \overline{\operatorname{co}}\left( \bar{Z}\right) \) . Then \( x \) is the barycenter of a regular Borel probability measure \( \mu \) that lives on \( \bar{Z} \) (Theorem 2). We can extend \( \mu \) to all of \( K \) by making \( \mu \left( B\right) = \mu \left( {B ... | Yes |
Lemma 6. Let \( K \) be a nonempty compact convex metrizable subset of \( E \) . Then \( C\left( K\right) \) contains a strictly convex member. | Proof. The metrizability of \( K \) ensures the separability of \( C\left( K\right) \) and hence that of \( A\left( K\right) \) . Let \( \left( {h}_{n}\right) \) be a dense sequence in \( {S}_{A\left( K\right) } \) ; define \( h = \mathop{\sum }\limits_{n}{h}_{n}^{2}/{2}^{n} \) . The \( M \) -test assures us that \( h ... | Yes |
Theorem 7. Let \( X \) be a separable Banach space with separable dual \( {X}^{ * } \) . Then the identity map \( {\operatorname{id}}_{K} \) on \( K \) is weak*-norm continuous at a weak* dense \( {\mathcal{G}}_{\delta } \) set of points of \( K \) whenever \( K \) is a weak* compact subset of \( {X}^{ * } \) . | Proof. For each \( \varepsilon > 0 \) let \( {A}_{\varepsilon } \) be the union of all \( W \cap K \), where \( W \) is a weak* open set in \( {X}^{ * } \) for which the norm diameter of \( W \cap K \) is \( \leq \varepsilon \) . Plainly each \( {A}_{e} \) is weak* open in \( K \) . Moreover, the points of weak*-norm c... | Yes |
Theorem 12. Let \( X \) be a separable real Banach space of infinite dimension whose dual unit ball has but countably many extreme points. Then \( X \) contains an isomorph of \( {c}_{0} \) . | Proof. Suppose ext \( {B}_{{X}^{ * }} = {\left\{ \pm {x}_{n}^{ * }\right\} }_{n \geq 1} \) and let \( 1 > {\varepsilon }_{1} > {\varepsilon }_{2} > \cdots > {\varepsilon }_{n} \downarrow 0 \) . Define a new norm on \( X \) by\n\n\[ \parallel \left| x\right| \parallel = \sup \left\{ {\left| {{x}^{ * }x}\right| : {x}^{ *... | Yes |
Lemma 1. If \( \left| x\right| = 1 = \left| y\right| \), then \( \left( {{\varphi }_{x},{\varphi }_{y}}\right) = x \cdot y \) . | Proof. Since \( \left| x\right| = 1 = \left| y\right| \), there is a vector \( {y}^{\prime } \) orthogonal to \( y \) such that\n\n\[ x = \left( {x \cdot y}\right) y + {y}^{\prime }.\]\n\nFor this \( {y}^{\prime } \) and all \( z \) we have\n\n\[ x \cdot z = \left( {x \cdot y}\right) \left( {y \cdot z}\right) + {y}^{\p... | Yes |
Theorem 3 (Lindenstrauss-Pelczynski). Let \( \\left( {x}_{n}\\right) \) be a normalized unconditional basis for \( {l}_{1} \). Then \( \\left( {x}_{n}\\right) \) is equivalent to the unit vector basis. | Proof. Suppose \( K > 0 \) is chosen so that\n\n\[ \n\\begin{Vmatrix}{\\mathop{\\sum }\\limits_{n}{b}_{n}{a}_{n}{x}_{n}}\\end{Vmatrix} \\leq K\\begin{Vmatrix}{\\mathop{\\sum }\\limits_{n}{a}_{n}{x}_{n}}\\end{Vmatrix}\n\]\n\nholds for any \( \\left( {b}_{n}\\right) \\in {B}_{{l}_{\\infty }} \) and any sequence of scalar... | Yes |
Lemma 1. Let \( A \in {\mathcal{P}}_{ < \infty }\left( \mathbb{N}\right) \) and \( B \in {\mathcal{P}}_{\infty }\left( \mathbb{N}\right) \). 1. Suppose \( B \) accepts \( A \). Then each \( C \in {\mathcal{P}}_{\infty }\left( B\right) \) also accepts \( A \). 2. Suppose \( B \) rejects \( A \). Then each \( C \in {\mat... | Proof. Part 1 is immediate from the fact that whenever \( C \in {\mathcal{P}}_{\infty }\left( B\right) \), \( {\mathcal{P}}_{\infty }\left( {A, C}\right) \subseteq {\mathcal{P}}_{\infty }\left( {A, B}\right) \n\nPart 2 is clear, too, since were there a \( C \in {\mathcal{P}}_{\infty }\left( B\right) \) that did not rej... | Yes |
Lemma 3. Suppose \( Z \in {\mathcal{P}}_{\infty }\left( \mathbb{N}\right) \) accepts or rejects each of its finite subsets (on an individual basis).\n\n1. If \( A \in {\mathcal{P}}_{ < \infty }\left( Z\right) \) and \( Z \) rejects \( A \), then \( Z \) rejects \( A \cup \{ n\} \) for all but finitely many \( n \in Z \... | Proof. 1. By hypothesis, \( Z \) accepts any \( A \cup \{ n\} \left( {n \in Z}\right) \) it does not reject; hence, were part 1 to fail,\n\n\[ B = \{ n \in Z : Z\text{ accepts }A \cup \{ n\} \} \in {\mathcal{P}}_{\infty }\left( Z\right) . \]\n\nConsider \( {\mathcal{P}}_{\infty }\left( {A, B}\right) \) . If \( X \in {\... | Yes |
Lemma 4. Every \( \tau \) -open set in \( {\mathcal{P}}_{\infty }\left( \mathbb{N}\right) \) is a Ramsey collection. | Proof. Let \( \mathcal{S} \) be a \( \tau \) -open subset of \( {\mathcal{P}}_{\infty }\left( \mathbb{N}\right) \) . By Lemma 2, there is a \( Z \in {\mathcal{P}}_{\infty }\left( \mathbb{N}\right) \) that accepts or rejects each of its own finite subsets on an individual basis, acceptance (and rejection) being relative... | Yes |
Lemma 6. Let \( T \) be a topological space. Then the collection of subsets of \( T \) having the Baire property forms a \( \sigma \) -algebra containing the open sets of \( T \) . | Proof. First of all, if a set has the Baire property, so does its complement. To see this, notice that closed sets have the Baire property differing as they do from their interior by their nowhere dense boundary. Notice too that any set with a meager symmetric difference from another having the Baire property has the B... | Yes |
Proposition 1. Let \( \Omega \) be a set, \( \left( {\Omega }_{n}\right) \) be a tree of subsets of \( \Omega, B \) be a bounded subset of \( {l}_{\infty }\left( \Omega \right) \) and \( \delta > 0 \) . Suppose we have a (Rademacher-like) sequence \( \left( {b}_{n}\right) \) in \( B \) such that whenever \( {2}^{n - 1}... | Proof. Let \( {t}_{1},\ldots ,{t}_{n} \) be real numbers. We look at \( {\begin{Vmatrix}\mathop{\sum }\limits_{{j = 1}}^{n}{t}_{j}{b}_{j}\end{Vmatrix}}_{\infty } \) ; its biggest of possible values is \( \left( {\mathop{\sup }\limits_{B}\parallel b\parallel }\right) \mathop{\sum }\limits_{{j = 1}}^{n}\left| {t}_{j}\rig... | No |
Lemma 1. The class of Banach spaces having weak* sequentially compact dual ball is closed under the following operations:\n\n1. Taking dense continuous linear images\n\n2. Quotients\n\n3. Subspaces | Proof. If \( T : X \rightarrow Y \) is a bounded linear operator with dense range, then \( {T}^{ * } : {Y}^{ * } \rightarrow {\dot{X}}^{ * } \) is a bounded linear operator that’s one to one. It follows that \( {T}^{ * } \) is a weak* homeomorphism between \( {B}_{{Y}^{ * }} \) and \( {T}^{ * }{B}_{{Y}^{ * }} \) . This... | No |
Lemma 2 (A. Grothendieck). Let \( K \) be a weakly closed subset of the Banach space \( X \) . Suppose that for each \( \varepsilon > 0 \) there is a weakly compact set \( {K}_{e} \) in \( X \) such that\n\n\[ K \subseteq {K}_{\varepsilon } + \varepsilon {B}_{X} \]\n\nThen \( K \) is weakly compact. | Proof. Let \( {\bar{K}}^{\text{weak } * } \) denote \( K \) ’s weak* closure up in \( {X}^{* * } \) . If \( {\bar{K}}^{\text{weak } * } \) should find itself back in \( X \), then we are done. In fact, \( K \), sitting as it does in \( {K}_{1} + {B}_{X} \) for some weakly compact set \( {K}_{1} \) corresponding to \( \... | Yes |
Theorem 4 (D. Amir, J. Lindenstrauss). Any subspace of a weakly compactly generated Banach space has a weak* sequentially compact dual ball. | Proof. Suppose first that \( X \) is a weakly compactly generated Banach space and assume that \( K \) is a weakly compact absolutely convex set in \( X \), whose linear span is dense in \( X \) . Let \( C \) be the weakly compact absolutely convex set produced in Lemma 3. The linear operator \( S : {X}_{C} \rightarrow... | No |
Lemma 5. Let \( \left( {x}_{n}^{ * }\right) \) be a sequence in \( {B}_{{X}^{ * }} \) with no weak* convergent subsequence. Then for any subsequence \( M \) of \( \mathbb{N} \) | \[ 0 < \delta \left( M\right) = \mathop{\sup }\limits_{{{y}^{ * },{z}^{ * }}}\inf \left| {{y}^{ * }\left( x\right) - {z}^{ * }\left( x\right) }\right| ,\] where the supremum is taken over all the subsequences \( {M}_{0} \) and \( {M}_{1} \) of \( M \) and all \( {y}^{ * } \in {\widetilde{\operatorname{co}}}^{ * }\left\... | No |
Lemma 7. Suppose \( X \) is a subspace of the Banach space \( Y \) . Identify \( {X}^{* * } \) with the subspace \( {X}^{ \bot \bot } \) in \( {Y}^{* * } \) . Let \( G \in {X}^{* * } \) be a Baire-1 member of \( {Y}^{* * } \) . Then \( G \) is a Baire-1 member of \( {X}^{* * } \) ; in fact, if \( \parallel G\parallel =... | Proof. Let \( {y}_{n} \in Y \) be chosen so that \( G = {\operatorname{weak}}^{ * }\mathop{\lim }\limits_{n}{y}_{n} \) . We claim that distance \( \left( {{B}_{X},\overline{\mathrm{{co}}}\left\{ {{y}_{n},{y}_{n + 1},\ldots }\right\} }\right) = 0 \) for all \( n \) . In fact, were this not so then there would be an \( n... | Yes |
If \( K \) is a compact Hausdorff space and \( f : K \rightarrow R \) is a bounded function with no points of continuity, then there exist a nonempty closed subset \( L \) of \( K \) and real numbers \( r,\delta \) with \( \delta > 0 \) so that\n\n(*) For every nonempty relatively open subset \( \cup \) of \( L \) ther... | Proof. For each \( n \) let\n\n\[ \n{C}_{n} = \left\{ {x \in K : \text{ if }U\text{ is open and contains }x,\text{ there are }y, z \in U\text{ with }}\right.\n\]\n\n\[ \nf\left( y\right) - f\left( z\right) > \left. \frac{1}{n}\right\} .\n\]\n\nSince \( f \) is nowhere continuous, \( K = { \cup }_{n}{C}_{n} \) . It is e... | Yes |
Lemma 9. Let \( L \) be a compact Hausdorff space and \( f : L \rightarrow R \) be a bounded function. Suppose \( r,\delta \) are real numbers with \( \delta > 0 \) and assume that\n\n(*) For each nonempty relatively open subset \( U \) of \( L \) there are \( y, z \in U \) with \( f\left( z\right) < r \) and \( f\left... | Proof. By (*) there are \( {y}_{1},{y}_{2} \in L \) with \( f\left( {y}_{1}\right) > r + \delta \) and \( f\left( {y}_{2}\right) < r \) .\n\nChoose \( {g}_{1} \in \mathcal{G} \) so that \( {g}_{1}\left( {y}_{1}\right) > r + \delta \) and \( {g}_{1}\left( {y}_{2}\right) < r \) . Consider the nonempty open subsets \( {A}... | No |
Theorem 10 (Odell-Rosenthal). Let \( X \) be a separable Banach space. Then the following are equivalent:\n\n1. \( X \) contains no isomorph of \( {l}_{1} \) .\n\n2. \( {B}_{X} \) is weak* sequentially dense in \( {B}_{{X}^{* * }} \) .\n\n3. \( {B}_{{X}^{* * }} \) is weak* sequentially compact. | Proof. Suppose \( {x}^{* * } \in {B}_{{X}^{* * }} \) is not the weak* limit of any sequence of terms from \( X \) . Since \( X \) is separable, \( {B}_{{X}^{ * }} \) is weak* compact, weak* metrizable; therefore, our Basic Lemma tells us that \( {\left. {x}^{* * }\right| }_{\left( {B}_{X}*,{\text{weak }}^{ * }\right) }... | No |
Lemma 2. Let \( \left( {x}_{n}\right) \) be a sequence in the Banach space \( X \) . For any \( K > 0 \), the set \[ {\mathcal{B}}_{K} = \left\{ {M = \left( {m}_{i}\right) \in {\mathcal{P}}_{\infty }\left( \mathbf{N}\right) : \mathop{\sup }\limits_{n}\begin{Vmatrix}{\mathop{\sum }\limits_{{i = 1}}^{n}{x}_{{m}_{i}}}\end... | Proof. We show that \( {\mathcal{P}}_{\infty }\left( \mathbf{N}\right) \smallsetminus {\mathcal{B}}_{K} \) is relatively open. Take an \( M = \left( {m}_{i}\right) \in \) \( {\mathcal{P}}_{\infty }\left( \mathbb{N}\right) \smallsetminus {\mathcal{B}}_{K} \) . There must be some \( n \) such that \[ \begin{Vmatrix}{\mat... | Yes |
Lemma 3. Suppose \( \left( {x}_{n}\right) \) is a bimonotone basic sequence in the Banach space \( X \) and for each \( K > 0 \) let\n\n\[ \n{\mathcal{B}}_{K} = \left\{ {M = \left( {m}_{i}\right) \in {\mathcal{P}}_{\infty }\left( \mathbf{N}\right) : \mathop{\sup }\limits_{n}\begin{Vmatrix}{\mathop{\sum }\limits_{{i = 1... | Proof. By Lemma 2, the set \( {\mathcal{B}}_{1} \) is relatively closed in \( {\mathcal{P}}_{\infty }\left( \mathbb{N}\right) \) ; therefore, \( {\mathcal{B}}_{1} \) is completely Ramsey. It follows that there is an \( {M}_{1} \in {\mathcal{P}}_{\infty }\left( M\right) \) for which\n\n\[ \n{\mathcal{B}}_{\infty }\left(... | Yes |
Lemma 5. Let \( \left( {x}_{n}\right) \) be a normalized basic sequence in \( X \) . Suppose that each subsequence of \( \left( {x}_{n}\right) \) has a subsequence \( \left( {y}_{n}\right) \) such that\n\n\[ \mathop{\sup }\limits_{n}\begin{Vmatrix}{\mathop{\sum }\limits_{{i = 1}}^{n}{\left( -1\right) }^{i}{y}_{i}}\end{... | Proof. Plainly the statement of the lemma allows us to equivalently renorm \( \left\lbrack {x}_{n}\right\rbrack \) to achieve our goal: locate a copy of \( {c}_{0} \) inside \( \left\lbrack {x}_{n}\right\rbrack \) . We do so to make \( \left( {x}_{n}\right) \) a bimonotone normalized basic sequence.\n\nNow looking at t... | Yes |
For a single toss of a coin the sample space \( \Omega \) consists of two points: | \[ \Omega = \{ \mathrm{H},\mathrm{T}\} \] where \( \mathrm{H} = \) \ | Yes |
Example 4 (Sampling with Replacement). This is an experiment in which at each step one ball is drawn at random and returned again. The balls are numbered \( 1,\ldots, M \), so that each sample of \( n \) balls can be presented in the form \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) , where \( {a}_{i} \) is the label ... | We prove this by induction. Let \( N\left( {M, n}\right) \) be the number of outcomes of interest. It is clear that when \( k \leq M \) we have\n\n\[ N\left( {k,1}\right) = k = {C}_{k}^{1}. \]\n\nNow suppose that \( N\left( {k, n}\right) = {C}_{k + n - 1}^{n} \) for \( k \leq M \) ; we will show that this formula conti... | Yes |
Suppose that \( n \leq M \) and that the selected balls are not returned. In this case we again consider two possibilities, namely ordered and unordered samples. | For ordered samples without replacement (called in combinatorics arrangements of \( n \) out of \( M \) elements without repetitions) the sample space\n\n\[ \Omega = \left\{ {\omega : \omega = \left( {{a}_{1},\ldots ,{a}_{n}}\right) ,{a}_{k} \neq {a}_{l}, k \neq l,{a}_{i} = 1,\ldots, M}\right\} ,\]\n\nconsists of \( M\... | Yes |
We consider the structure of the sample space in the problem of allocation of \( n \) objects (balls, etc.) among \( M \) cells (boxes, etc.). | Let the cells be numbered \( 1,2,\ldots, M \), and suppose first that the objects are distinguishable (numbered \( 1,2,\ldots, n \) ). Then an allocation of the \( n \) objects among the \( M \) cells is completely described by an (ordered) collection \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \), where \( {a}_{i} \) i... | No |
Example 7 (Coincidence Problem). Let an urn contain \( M \) balls numbered \( 1,2,\ldots, M \) . We draw an ordered sample of size \( n \) with replacement. It is clear that then\n\n\[ \Omega = \left\{ {\omega : \omega = \left( {{a}_{1},\ldots ,{a}_{n}}\right) ,{a}_{i} = 1,\ldots, M}\right\} \]\n\nand \( N\left( \Omega... | Clearly \( N\left( A\right) = M\left( {M - 1}\right) \cdots (M - n + 1) \), and therefore\n\n\[ \mathrm{P}\left( A\right) = \frac{{\left( M\right) }_{n}}{{M}^{n}} = \left( {1 - \frac{1}{M}}\right) \left( {1 - \frac{2}{M}}\right) \cdots \left( {1 - \frac{n - 1}{M}}\right) . \] | Yes |
Consider a lottery that is run in the following way. There are \( M \) tickets numbered \( 1,2,\ldots, M \), of which \( n \), numbered \( 1,\ldots, n \) , win prizes \( \left( {M \geq {2n}}\right) \) . You buy \( n \) tickets, and ask for the probability \( \left( {P\text{, say}}\right) \) of winning at least one priz... | Since the order in which the tickets are drawn plays no role in the presence or absence of winners in your purchase, we may suppose that the sample space has the form\n\n\[ \Omega = \left\{ {\omega : \omega = \left\lbrack {{a}_{1},\ldots ,{a}_{n}}\right\rbrack ,{a}_{k} \neq {a}_{l}, k \neq l,{a}_{i} = 1,\ldots, M}\righ... | Yes |
Consider a family with two children. We ask for the probability that both children are boys, assuming\n\n(a) that the older child is a boy;\n\n(b) that at least one of the children is a boy. | The sample space is\n\n\[ \Omega = \{ \mathrm{{BB}},\mathrm{{BG}},\mathrm{{GB}},\mathrm{{GG}}\} \]\n\nwhere BG means that the older child is a boy and the younger is a girl, etc.\n\nLet us suppose that all sample points are equally probable:\n\n\[ \mathrm{P}\left( \mathrm{{BB}}\right) = \mathrm{P}\left( \mathrm{{BG}}\r... | Yes |
Let an urn contain two coins: \( {A}_{1} \), a fair coin with probability \( \frac{1}{2} \) of falling \( \mathrm{H} \) ; and \( {\mathrm{A}}_{2} \), a biased coin with probability \( \frac{1}{3} \) of falling \( \mathrm{H} \) . A coin is drawn at random and tossed. Suppose that it falls head. We ask for the probabilit... | Let us construct the corresponding probabilistic model. Here it is natural to take the sample space to be the set \( \Omega = \left\{ {{A}_{1}\mathrm{H},{A}_{1}\mathrm{\;T},{A}_{2}\mathrm{H},{A}_{2}\mathrm{\;T}}\right\} \), which describes all possible outcomes of a selection and a toss \( \left( {{A}_{1}\mathrm{H}}\ri... | Yes |
Example 3. Let \( \xi \) be a Bernoulli random variable, taking the values 1 and 0 with probabilities \( p \) and \( q \) . Then | \[ \mathrm{E}\xi = 1 \cdot \mathrm{P}\{ \xi = 1\} + 0 \cdot \mathrm{P}\{ \xi = 0\} = p. \] | Yes |
Let \( {\xi }_{1},\ldots ,{\xi }_{n} \) be \( n \) Bernoulli random variables with \( \mathrm{P}\left\{ {{\xi }_{i} = 1}\right\} = \) \( p,\mathrm{P}\left\{ {{\xi }_{i} = 0}\right\} = q, p + q = 1 \). Then if \( {S}_{n} = {\xi }_{1} + \cdots + {\xi }_{n} \), we find that \( \mathrm{E}{S}_{n} = {np} \). | This result can also be obtained in a different way. It is easy to see that \( \mathrm{E}{S}_{n} \) is not changed if we assume that the Bernoulli random variables \( {\xi }_{1},\ldots ,{\xi }_{n} \) are independent. With this assumption, we have according to (4) \( \mathrm{P}\left( {{S}_{n} = k}\right) = {C}_{n}^{k}{p... | Yes |
If \( \xi \) is a Bernoulli random variable, taking the values 1 and 0 with probabilities \( p \) and \( q \), then | \[ \operatorname{Var}\xi = \mathrm{E}{\left( \xi - \mathrm{E}\xi \right) }^{2} = \mathrm{E}{\left( \xi - p\right) }^{2} = {\left( 1 - p\right) }^{2}p + {p}^{2}q = {pq}. \] | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.