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Proposition 6.1.3. The Haar system is a monotone basis in \( {L}_{p} \) for \( 1 \leq p < \infty \) . | Proof. Let us consider an increasing sequence of \( \sigma \) -algebras, \( {\left( {\mathcal{B}}_{n}\right) }_{n = 1}^{\infty } \), contained in the Borel \( \sigma \) -algebra of \( \left\lbrack {0,1}\right\rbrack \) defined as follows: we let \( {\mathcal{B}}_{1} \) be the trivial \( \sigma \) -algebra, \( \{ \varno... | Yes |
Lemma 6.1.5. Suppose \( p > 2 \) . Then\n\n\[ \frac{{p}^{p - 2}}{{\left( p - 1\right) }^{p - 1}} < 1 \] | Proof. If we let \( t = p - 1 \), inequality (6.1) is equivalent to\n\n\[ H\left( t\right) = - \left( {t - 1}\right) \log \left( {1 + t}\right) + t\log \left( t\right) > 0,\;\forall t > 1. \]\n\nIndeed, differenting \( H \) gives\n\n\[ {H}^{\prime }\left( t\right) = \frac{2}{t + 1} - \log \left( \frac{1 + t}{t}\right) ... | Yes |
(a) The following inequality holds for all \( \\left( {x, y}\\right) \) with \( x \\geq 0 \) and \( y \\geq 0 \) :\n\n\[ \n\\frac{{\\left( p - 1\\right) }^{p - 1}}{{p}^{p - 2}}\\varphi \\left( {x, y}\\right) \\leq {\\left( p - 1\\right) }^{p}{x}^{p} - {y}^{p}.\n\] | Proof. (a) By homogeneity we can suppose that \( x + y = 1 \) . Then it suffices to show that the function\n\n\[ \nG\\left( x\\right) = \\frac{{p}^{p - 2}}{{\\left( p - 1\\right) }^{p - 1}}\\left( {{\\left( p - 1\\right) }^{p}{x}^{p} - {\\left( 1 - x\\right) }^{p}}\\right) - {px} + 1,\;0 \\leq x \\leq 1,\n\]\n\nis nonn... | Yes |
Theorem 6.2.2 (Khintchine’s Inequalities). For every \( 1 \leq p < \infty \) there exist positive constants \( {A}_{p} \) and \( {B}_{p} \) such that for every finite sequence of scalars \( {\left( {a}_{i}\right) }_{i = 1}^{n} \) and \( n \in \mathbb{N} \) we have\n\n\[ \n{A}_{p}{\left( \mathop{\sum }\limits_{{i = 1}}^... | We will not prove this here, but it will be derived as a consequence of a more general result below. | No |
Theorem 6.2.5 (Kahane-Khintchine Inequalities). For each \( 1 \leq p < \infty \) there exists a constant \( {C}_{p} \) such that for every Banach space \( X \) and finite sequence \( {\left( {x}_{i}\right) }_{i = 1}^{n} \) in \( X \), the following inequality holds:\n\n\[ \mathbb{E}\begin{Vmatrix}{\mathop{\sum }\limits... | We will prove the Kahane-Khintchine inequalities (and this will imply the Khintchine inequalities by taking \( X = \mathbb{R} \) or \( X = \mathbb{C} \) ), but first we shall establish three lemmas on our way to the proof. To avoid repetitions, in all three lemmas, \( \left( {\Omega ,\sum ,\mathbb{P}}\right) \) will be... | Yes |
Lemma 6.2.6. Let \( f : \Omega \rightarrow X \) be a symmetric random variable. Then for all \( x \in X \) we have\n\n\[ \mathbb{P}\left( {\parallel f + x\parallel \geq \parallel x\parallel }\right) \geq \frac{1}{2} \] | Proof. Let us take any \( x \in X \) . For every \( \omega \in \Omega \), using the convexity of the norm of \( X \), clearly \( \parallel f\left( \omega \right) + x\parallel + \parallel x - f\left( \omega \right) \parallel \geq 2\parallel x\parallel \) . Then, either \( \parallel f\left( \omega \right) + x\parallel \g... | Yes |
Lemma 6.2.7. For all \( \lambda > 0 \) ,\n\n\[ \mathbb{P}\left( {\mathop{\max }\limits_{{m \leq n}}\begin{Vmatrix}{\Lambda }_{m}\end{Vmatrix} > \lambda }\right) \leq 2\mathbb{P}\left( {\begin{Vmatrix}{\Lambda }_{n}\end{Vmatrix} > \lambda }\right) . \] | Proof. Given \( \lambda > 0 \), for \( m = 1,\ldots, n \) put\n\n\[ {\Omega }_{m}^{\left( \lambda \right) } = \left\{ {\omega \in \Omega : \begin{Vmatrix}{{\Lambda }_{m}\left( \omega \right) }\end{Vmatrix} > \lambda \text{ and }\begin{Vmatrix}{{\Lambda }_{j}\left( \omega \right) }\end{Vmatrix} \leq \lambda \text{ for a... | Yes |
Lemma 6.2.8. For all \( \lambda > 0 \) , \[ \mathbb{P}\left( {\begin{Vmatrix}{\Lambda }_{n}\end{Vmatrix} > {2\lambda }}\right) \leq 4{\left( \mathbb{P}\left( \begin{Vmatrix}{\Lambda }_{n}\end{Vmatrix} > \lambda \right) \right) }^{2}. \] | Proof. We will keep the notation that we introduced in the previous lemma. Notice that for each \( 1 \leq m \leq n \), the random variable \( \begin{Vmatrix}{\mathop{\sum }\limits_{{i = m}}^{n}{\varepsilon }_{i}{x}_{i}}\end{Vmatrix} \) is independent of each of \( {\varepsilon }_{1},\ldots ,{\varepsilon }_{m} \), and h... | Yes |
Proposition 6.2.9 (Generalized Parallelogram Law). Suppose that \( H \) is a Hilbert space. Then for every finite sequence \( {\left( {x}_{i}\right) }_{i}^{n} \) in \( H \) , | \[ \mathbb{E}{\begin{Vmatrix}\mathop{\sum }\limits_{{i = 1}}^{n}{\varepsilon }_{i}{x}_{i}\end{Vmatrix}}^{2} = \mathop{\sum }\limits_{{i = 1}}^{n}{\begin{Vmatrix}{x}_{i}\end{Vmatrix}}^{2} \] Proof. For any vectors \( {\left\{ {x}_{i}\right\} }_{i = 1}^{n} \) in \( H \) we have \[ \mathbb{E}{\begin{Vmatrix}\mathop{\sum }... | Yes |
Proposition 6.2.12. If a Banach space \( X \) has type \( p \), then \( {X}^{ * } \) has cotype \( q \), where \( \frac{1}{p} + \frac{1}{q} = 1 \), and \( {C}_{q}\left( {X}^{ * }\right) \leq {T}_{p}\left( X\right) \) . | Proof. Let us pick an arbitrary finite set \( {\left\{ {x}_{i}^{ * }\right\} }_{i = 1}^{n} \) in \( {X}^{ * } \) . Given \( \epsilon > 0 \), we can find \( {x}_{1},\ldots ,{x}_{n} \) in \( X \) such that \( \begin{Vmatrix}{x}_{i}\end{Vmatrix} = 1 \) and \( \left| {{x}_{i}^{ * }\left( {x}_{i}\right) }\right| \geq \left(... | Yes |
Theorem 6.2.13. Let \( 1 \leq p < \infty \) . For every finite set of functions \( {\left\{ {f}_{i}\right\} }_{i = 1}^{n} \) in \( {L}_{p}\left( \mu \right) \) , \[ {A}_{p}{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{i = 1}}^{n}{\left| {f}_{i}\right| }^{2}\right) }^{\frac{1}{2}}\end{Vmatrix}}_{p} \leq {\left( \mathb... | Proof. For each \( \omega \in \Omega \), from Khintchine’s inequalities, \[ {A}_{p}{\left( \mathop{\sum }\limits_{{i = 1}}^{n}{\left| {f}_{i}\left( \omega \right) \right| }^{2}\right) }^{1/2} \leq {\left( \mathbb{E}{\left| \mathop{\sum }\limits_{{i = 1}}^{n}{\varepsilon }_{i}{f}_{i}\left( \omega \right) \right| }^{p}\r... | Yes |
Theorem 6.2.14. (a) If \( 1 \leq p \leq 2,{L}_{p}\left( \mu \right) \) has type \( p \) and cotype 2. | Proof. (a) Let us prove first that if \( 1 \leq p \leq 2 \), then \( {L}_{p}\left( \mu \right) \) has type \( p \) . We recall this elementary inequality: | No |
Lemma 6.2.16. Let \( 0 < r < 1 \) . Then\n\n\[ \parallel f + g{\parallel }_{r} \geq \parallel f{\parallel }_{r} + \parallel g{\parallel }_{r} \]\n\nwhenever \( f \) and \( g \) are nonnegative functions in \( {L}_{r}\left( \mu \right) \) . | Proof. Without loss of generality we can assume that \( \parallel f + g{\parallel }_{r} = 1 \), and so \( {d\nu } = \) \( {\left( f + g\right) }^{r}{d\mu } \) is a probability measure. This implies\n\n\[ \parallel f{\parallel }_{r} = {\left( {\int }_{\Omega }{f}^{r}d\mu \right) }^{1/r} = {\left( {\int }_{\{ f + g > 0\}... | Yes |
Proposition 6.3.1. The Haar basis is not unconditional in \( {L}_{1} \) . | Proof. For each \( N \in \mathbb{N} \) let\n\n\[ \n{f}_{N}\left( t\right) = {2}^{1 - {2N}}{\chi }_{\left\lbrack 0,{2}^{1 - {2N}}\right\rbrack }\left( t\right) ,\;t \in \left\lbrack {0,1}\right\rbrack .\n\]\n\nLet us use the device of labeling the elements of the Haar system by their supports. With this convention, expa... | Yes |
Lemma 6.3.2. For every \( f \in {L}_{1} \) we have\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\int }_{0}^{1}f\left( t\right) {r}_{n}\left( t\right) {dt} = 0 \]\n\nThus \( {\left( f{r}_{n}\right) }_{n = 1}^{\infty } \) is weakly null for every \( f \in {L}_{1} \) . | Proof. The sequence \( {\left( {r}_{n}\right) }_{n = 1}^{\infty } \) is orthonormal in \( {L}_{2} \), which implies (by Bessel’s inequality) that for \( f \in {L}_{2} \), \n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\int }_{0}^{1}f\left( t\right) {r}_{n}\left( t\right) {dt} = 0 \]\n\nSince \( {\left( {r}_{n}\... | Yes |
Lemma 6.3.4. Let \( X \) be a Banach space such that \( {X}^{ * } \) is separable. Assume that \( K \) is a weak* compact set in \( {X}^{ * } \). Then \( K \) has a point of weak*-to-norm continuity. That is, there is \( {x}^{ * } \in K \) such that whenever a sequence \( {\left( {x}_{n}^{ * }\right) }_{n = 1}^{\infty ... | Proof. Let \( {\left( {\epsilon }_{n}\right) }_{n = 1}^{\infty } \) be a sequence of scalars converging to zero. Using that \( {X}^{ * } \) is separable for the norm topology, for each \( n \) there is a sequence of points \( {\left( {x}_{k, n}^{ * }\right) }_{k = 1}^{\infty } \subset \) \( {X}^{ * } \) such that\n\n\[... | Yes |
Lemma 6.3.5. Suppose \( X \) is a Banach space that embeds in a separable dual space. Then every closed bounded subset \( F \) of \( X \) has a point of weak-to-norm continuity. | Proof. Let \( F \) be a closed bounded subset of \( X \) . Suppose \( T : X \rightarrow {Y}^{ * } \) is an embedding in \( {Y}^{ * } \), where \( Y \) is a Banach space with separable dual. We can assume that \( \parallel x\parallel \leq \parallel {Tx}\parallel \leq M\parallel x\parallel \) for \( x \in X \), where \( ... | Yes |
Lemma 6.3.6. Suppose \( X \) is a Banach space that embeds in a separable dual space and let \( x \in {B}_{X} \) be a point of weak-to-norm continuity. If \( \left( {x}_{n}\right) \) is a weakly null sequence in \( X \) such that \( \lim \sup \begin{Vmatrix}{x + {x}_{n}}\end{Vmatrix} \leq 1 \), then \( \mathop{\lim }\l... | Proof. Put\n\n\[ \n{u}_{n} = \left\{ \begin{array}{ll} x + {x}_{n} & \text{ if }\begin{Vmatrix}{x + {x}_{n}}\end{Vmatrix} \leq 1, \\ \frac{x + {x}_{n}}{\begin{Vmatrix}x + {x}_{n}\end{Vmatrix}} & \text{ if }\begin{Vmatrix}{x + {x}_{n}}\end{Vmatrix} > 1, \end{array}\right. \]\n\nand observe that\n\n\[ \n{u}_{n} - x = {x}... | Yes |
Corollary 6.3.9. The space \( {L}_{1} \) does not have a boundedly complete basis. | Proof. We need only recall that by Theorem 3.2.15, a space with a boundedly complete basis is (isomorphic to) a separable dual space. | Yes |
Proposition 6.3.10. There is a norm-one linear projection \( P : {L}_{1}^{* * } \rightarrow {L}_{1} \) . | Proof. Let \( J \) be the cannonical embedding of \( {L}_{1} \) into \( \mathcal{M}\left\lbrack {0,1}\right\rbrack \) . Define \( Q : \mathcal{M}\left\lbrack {0,1}\right\rbrack \rightarrow {L}_{1} \) by \( Q\left( \mu \right) = f \), where \( {d\mu } = {fdm} + {d\nu } \) and \( v \) is singular with respect to the Lebe... | Yes |
Proposition 6.4.1. Let \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) be a sequence of norm-one, disjointly supported functions in \( {L}_{p} \) . Then \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) is a complemented basic sequence isometrically equivalent to the canonical basis of \( {\ell }_{p} \) . | Proof. The case \( p = 1 \) was seen in Lemma 5.1.1. Let us fix \( 1 < p < \infty \) . For every sequence of scalars \( {\left( {a}_{i}\right) }_{i = 1}^{\infty } \in {c}_{00} \), by the disjointness of the \( {f}_{i} \) ’s we have\n\n\[ \n{\begin{Vmatrix}\mathop{\sum }\limits_{{i = 1}}^{\infty }{a}_{i}{f}_{i}\end{Vmat... | Yes |
Proposition 6.4.3. If \( {\ell }_{q} \) embeds in \( {L}_{p} \), then either \( p \leq q \leq 2 \) or \( 2 \leq q \leq p \) . | Proof. Let us start by noticing that if \( {\left( {e}_{i}\right) }_{i = 1}^{\infty } \) is the canonical basis of \( {\ell }_{q} \), then for each \( n \) we have\n\n\[ \n\mathbb{E}{\begin{Vmatrix}\mathop{\sum }\limits_{{i = 1}}^{n}{\varepsilon }_{i}{e}_{i}\end{Vmatrix}}_{q} = {n}^{1/q} \n\]\n\nIf \( {\ell }_{q} \) em... | Yes |
Proposition 6.4.5. Suppose \( \left( {\Omega ,\sum ,\mu }\right) \) is a probability measure space and let \( 1 \leq \) \( p < \infty \) . Suppose \( X \) is an infinite-dimensional closed subspace of \( {L}_{p}\left( \mu \right) \) . Then the following are equivalent:\n\n(i) \( X \) is strongly embedded in \( {L}_{p}\... | Proof. Let us suppose that \( X \) is strongly embedded in \( {L}_{p}\left( \mu \right) \) but (ii) fails. Then there would exist a sequence \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) in \( X \) such that \( {\begin{Vmatrix}{f}_{n}\end{Vmatrix}}_{p} = 1 \) and \( {\begin{Vmatrix}{f}_{n}\end{Vmatrix}}_{q} \rightar... | Yes |
For each \( 1 \leq p < \infty \) the closed subspace spanned in \( {L}_{p} \) by the Rademacher functions, \( {R}_{p} \), is strongly embedded in \( {L}_{p} \) | since using Khintchine’s inequality, the \( {L}_{q} \) -norm and the \( {L}_{p} \) -norm are equivalent in \( {R}_{p} \) for all \( 1 \leq q < \infty \) | Yes |
Theorem 6.4.7. Suppose that \( X \) is an infinite-dimensional closed subspace of \( {L}_{p} \) for some \( 1 \leq p < \infty \) . If \( X \) is not strongly embedded in \( {L}_{p} \), then \( X \) contains a subspace isomorphic to \( {\ell }_{p} \) and complemented in \( {L}_{p} \) . | Proof. If \( X \) is not strongly embedded in \( {L}_{p} \), there is a sequence \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) in \( X \) with \( {\begin{Vmatrix}{f}_{n}\end{Vmatrix}}_{p} = 1 \) for all \( n \) such that \( {f}_{n} \rightarrow 0 \) almost everywhere. By Lemma 5.2.1 there exist a subsequence \( {\lef... | Yes |
Theorem 6.4.8 (Kadets and Pelczyński [147]). Suppose that \( X \) is an infinite-dimensional closed subspace of \( {L}_{p} \) for some \( 2 < p < \infty \) . Then the following are equivalent:\n\n(i) The space \( {\ell }_{p} \) does not embed in \( X \) .\n\n(ii) The space \( {\ell }_{p} \) does not embed complementabl... | Proof. \( \left( i\right) \Rightarrow \left( {ii}\right) \) and \( \left( {iv}\right) \Rightarrow \left( v\right) \) are obvious, and \( \left( {ii}\right) \Rightarrow \left( {iii}\right) \) was proved in Theorem 6.4.7. Let us complete the circle by showing that (iii) \( \Rightarrow \) (iv) and that \( \left( v\right) ... | Yes |
Proposition 6.4.11. If \( g \) is a Gaussian on some probability measure space \( \left( {\Omega ,\sum ,\mu }\right) \), then \( g \in {L}_{p}\left( \mu \right) \) for every \( 1 \leq p < \infty \) . | Proof. This is because\n\n\[ \n{\int }_{\Omega }{\left| g\left( \omega \right) \right| }^{p}{d\omega } = \frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{\left| x\right| }^{p}{e}^{-\frac{1}{2}{x}^{2}}{dx} \n\]\n\nand the last integral is finite and indeed computable in terms of the \( \Gamma \) function as\n\n\[ \n\f... | Yes |
Proposition 6.4.12. The space \( {\ell }_{2} \) embeds isometrically in \( {L}_{p} \) for all \( 1 \leq p < \infty \) . | Proof. Take \( {\left( {g}_{j}\right) }_{j = 1}^{\infty } \), a sequence of independent Gaussians on \( \left\lbrack {0,1}\right\rbrack \) . By Proposition 6.4.11, \( {\left( {g}_{j}\right) }_{j = 1}^{\infty } \subset {L}_{p} \) . We will show that \( \left\lbrack {g}_{j}\right\rbrack \) is isometrically isomorphic to ... | Yes |
Theorem 6.4.16. Let \( f \) be a p-stable random variable on a probability measure space \( \left( {\Omega ,\sum ,\mu }\right) \) for some \( 0 < p < 2 \) . Then\n\n(i) \( f \in {L}_{q}\left( \mu \right) \) for all \( 0 < q < p \) ;\n\n(ii) \( f \notin {L}_{p}\left( \mu \right) \) . | Proof. Suppose that \( f \) is normalized \( p \) -stable for some \( 0 < p < 2 \) with distribution of probability \( {\mu }_{p} \) . Then\n\n\[{\int }_{\Omega }{\left| f\left( \omega \right) \right| }^{q}{d\omega } = {\int }_{-\infty }^{\infty }{\left| x\right| }^{q}d{\mu }_{p}\left( x\right)\]\n\nFor every \( x \in ... | Yes |
Theorem 6.4.17. If \( 1 \leq p < 2 \) and \( p \leq q \leq 2 \), then \( {\ell }_{q} \) embeds isometrically in \( {L}_{p} \) . | Proof. We have already seen the cases \( q = p \) and \( q = 2 \) . For \( 1 \leq p < q < 2 \), let \( {\left( {f}_{j}\right) }_{j = 1}^{\infty } \) be a sequence of independent normalized \( q \) -stable random variables on \( \left\lbrack {0,1}\right\rbrack \) . Then we can repeat the argument we used in Proposition ... | Yes |
Theorem 6.4.20. Let \( 1 < p, q < \infty \) . Then \( {\ell }_{q} \) embeds complementably in \( {L}_{p} \) if and only if \( q = p \) or \( q = 2 \) . | Proof. We know (Propositions 6.4.2 and 6.4.1) that both \( q = p \) and \( q = 2 \) allow complemented embeddings. Suppose \( {\ell }_{q} \) embeds in \( {L}_{p} \) complementably and \( q \notin \) \( \{ 2, p\} \) . By Theorem 6.4.18 we must have \( p < q < 2 \) . Taking duals, it follows that \( {\ell }_{{q}^{\prime ... | Yes |
Theorem 7.1.2. Let \( \mu \) be a \( \sigma \) -finite measure on some measurable space \( \left( {\Omega ,\sum }\right) \) . Suppose that \( T \) is an operator from a Banach space \( X \) into \( {L}_{p}\left( \mu \right) \) and that \( 1 \leq p < \) \( q < \infty \) . Suppose \( 0 < C < \infty \) . Then the followin... | Proof. \( \left( a\right) \Rightarrow \left( b\right) \) Since \( \left( {\Omega ,{hd\mu }}\right) \) is a probability measure space and \( p < q \), the \( {L}_{p}\left( {hd\mu }\right) \) -norm is smaller than the \( {L}_{q}\left( {hd\mu }\right) \) -norm, and thus we have\n\n\[{\left( {\int }_{\Omega }{\left( \matho... | Yes |
Lemma 7.1.3. Let \( r = q/p > 1 \) . Given \( {f}_{1},\ldots ,{f}_{n} \in {W}_{0} \) [respectively, \( W \) ] and \( {c}_{1},\ldots ,{c}_{n} \geq 0 \) with \( {c}_{1} + \cdots + {c}_{n} \leq 1 \), then \( {\left( {c}_{1}{f}_{1}^{r} + \cdots + {c}_{n}{f}_{n}^{r}\right) }^{1/r} \in {W}_{0} \) [respectively, W]. | Proof. It suffices to consider the case of \( {W}_{0} \) . Suppose\n\n\[ 0 \leq {f}_{k} \leq {\left( \mathop{\sum }\limits_{{j = 1}}^{{m}_{k}}{\left| T{x}_{jk}\right| }^{q}\right) }^{p/q},\;1 \leq k \leq n, \]\n\nwhere \( \mathop{\sum }\limits_{{j = 1}}^{{m}_{k}}{\begin{Vmatrix}{x}_{jk}\end{Vmatrix}}^{q} \leq {C}^{-q} ... | Yes |
Theorem 7.1.4. Let \( 1 \leq p < \infty \) . Suppose that \( T \) is an operator from a Banach space \( X \) into \( {L}_{p}\left( \mu \right) \) . If \( X \) has type 2, then there exists a constant \( C = C\left( p\right) \) such that for every finite sequence \( {\left( {x}_{k}\right) }_{k = 1}^{n} \) in \( X \) we ... | Proof. By Theorem 6.2.13, for every \( 1 \leq p < \infty \) there is a constant \( c = c\left( p\right) \) such that for every finite set of vectors \( {\left( {x}_{k}\right) }_{k = 1}^{n} \) in \( X \) ,\n\n\[{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{k = 1}}^{n}{\left| T{x}_{k}\right| }^{2}\right) }^{\frac{1}{2}... | Yes |
Corollary 7.1.5. (a) Every operator from a subspace of \( {L}_{r}\left( \mu \right) \left( {2 \leq r < \infty }\right) \) into \( {L}_{p}\left( \mu \right) \left( {1 \leq p < 2}\right) \) factors through a Hilbert space. | Corollary 7.1.5 follows immediately from Theorems 7.1.2 and 7.1.4. | Yes |
Lemma 7.1.7. Let \( 1 \leq p < q < 2 \) . Suppose that \( \gamma = {\left( {\gamma }_{j}\right) }_{j = 1}^{\infty } \) is a sequence of independent normalized q-stable random variables. Then for every finite sequence of functions \( {\left( {f}_{j}\right) }_{j = 1}^{n} \) in \( {L}_{p}\left( \mu \right) \) , \[ {\begin... | Proof. We recall from Theorem 6.4.17 that there is a constant \( c = c\left( {p, q}\right) \) such that \[ {\left( \mathbb{E}{\left| \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{j}{\gamma }_{j}\right| }^{p}\right) }^{1/p} = {c}^{-1}{\left( \mathop{\sum }\limits_{{j = 1}}^{n}{\left| {a}_{j}\right| }^{q}\right) }^{1/q},\;{\le... | Yes |
If \( X \) is a Banach space that embeds in \( {L}_{p} \) for some \( 0 < p < 1 \) , does \( X \) embed in \( {L}_{1} \) ? | In the isometric setting the answer is negative: a Banach space that embeds isometrically in \( {L}_{p} \) for some \( 0 < p < 1 \) need not embed isometrically in \( {L}_{1} \), as Koldobsky proved in 1996 [173]; see also [161]. In the isomorphic case the only known result is that \( X \) embeds in \( {L}_{1} \) if an... | No |
Lemma 7.2.1. Suppose \( f, g \in {L}_{p}\left( {1 \leq p < \infty }\right) \) . Then if \( 0 < \theta < 1 \), we have \( {\left| f\right| }^{1 - \theta }{\left| g\right| }^{\theta } \in {L}_{p} \) and\n\n\[ \n{\begin{Vmatrix}{\left| f\right| }^{1 - \theta }{\left| g\right| }^{\theta }\end{Vmatrix}}_{p} \leq \parallel f... | Proof. Just note that for \( s, t \geq 0 \) we have \( {s}^{1 - \theta }{t}^{\theta } \leq \left( {1 - \theta }\right) s + {\theta t} \) . Then, assuming \( \parallel f{\parallel }_{p},\parallel g{\parallel }_{p} > 0 \), by convexity we have\n\n\[ \n{\begin{Vmatrix}{\left( \frac{\left| f\right| }{\parallel f{\parallel ... | Yes |
Lemma 7.2.4. Both parameters \( {\alpha }_{n}\left( X\right) \) and \( {\beta }_{n}\left( X\right) \) are submultiplicative, i.e.,\n\n\[ \n{\alpha }_{mn}\left( X\right) \leq {\alpha }_{m}\left( X\right) {\alpha }_{n}\left( X\right) ,\;m, n \in \mathbb{N}, \n\] \n\n(7.9) \n\nand \n\n\[ \n{\beta }_{mn}\left( X\right) \le... | Proof. Let us take \( m \times n \) vectors in the unit ball of \( X \) and consider them as a matrix \( {\left( {x}_{ij}\right) }_{i, j = 1}^{m, n} \) . Let \( {\left( {\varepsilon }_{ij}\right) }_{i, j = 1}^{m, n} \) be a Rademacher sequence, and \( {\left( {\varepsilon }_{i}^{\prime }\right) }_{i = 1}^{n} \) another... | Yes |
Proposition 7.2.5. Suppose \( p < 2 < q \) .\n\n(a) In order that \( X \) have type \( r \) for some \( p < r \) it is necessary and sufficient that for some \( N,{\alpha }_{N}\left( X\right) < {N}^{\frac{1}{p} - \frac{1}{2}} \) . | Proof. One easily checks that if \( X \) has type \( r > p \) [respectively, cotype \( s < q \) ], then \( {\alpha }_{N}\left( X\right) < {N}^{\frac{1}{p} - \frac{1}{2}} \) [respectively, \( {\beta }_{N}\left( X\right) < {N}^{\frac{1}{2} - \frac{1}{q}} \) ] for some \( N \) by taking arbitrary sequences of vectors \( {... | Yes |
Theorem 7.3.2. Let \( T \) be an operator from a Banach space \( X \) into a Banach space \( Y \) . Suppose that there exist operators \( S : X \rightarrow H \) and \( R : H \rightarrow Y \) satisfying \( T = {RS} \) . If \( {\left( {x}_{j}\right) }_{j = 1}^{m} \) and \( {\left( {z}_{i}\right) }_{i = 1}^{n} \) are vect... | Proof. The proof easily follows from the comments we made. Indeed, given \( {x}_{1},\ldots ,{x}_{m} \) and \( {z}_{1},\ldots ,{z}_{n} \) in \( X \) satisfying (7.15), since the collections of vectors \( {\left( S{x}_{j}\right) }_{j = 1}^{m} \) and \( {\left( S{z}_{i}\right) }_{i = 1}^{n} \) lie inside \( H \), we have\... | Yes |
Proposition 7.3.3. Given \( n, m \in \mathbb{N} \) and any two sets of vectors \( {\left( {x}_{j}\right) }_{j = 1}^{m} \) and \( {\left( {z}_{i}\right) }_{i = 1}^{n} \) in a Banach space \( X \), the following are equivalent:\n\n(a) There is a real \( n \times m \) matrix \( A = \left( {a}_{ij}\right) \) such that \( \... | Proof. Assume that (a) holds. Then, since \( \parallel A{\parallel }_{{\ell }_{2}^{m} \rightarrow {\ell }_{2}^{n}} \leq 1 \), it follows that\n\n\[ \n\mathop{\sum }\limits_{{i = 1}}^{n}{\left| {x}^{ * }\left( {z}_{i}\right) \right| }^{2} = \mathop{\sum }\limits_{{i = 1}}^{n}{\left| {x}^{ * }\left( \mathop{\sum }\limits... | Yes |
Lemma 7.3.5. Let \( \mathcal{V} \) be a real vector space. Given two subsets \( \mathcal{A},\mathcal{B} \) of \( \mathcal{V} \) such that \( \mathcal{V} = \operatorname{cone}\left( \mathcal{B}\right) - \operatorname{cone}\left( \mathcal{A}\right) \), and two functions \( \phi : \mathcal{A} \rightarrow \mathbb{R},\psi :... | Proof. The implication \( \left( i\right) \Rightarrow \left( {ii}\right) \) is immediate.\n\n(ii) \( \Rightarrow \) (i) Let us define \( p : \mathcal{V} \rightarrow \lbrack - \infty ,\infty ) \) as\n\n\[ p\left( v\right) = \inf \left\{ {\mathop{\sum }\limits_{{j = 1}}^{n}{\beta }_{j}\psi \left( {b}_{j}\right) - \mathop... | Yes |
Lemma 7.4.3. Let \( X \) be a Banach space. Assume that the sets of vectors \( {\left\{ {z}_{i}\right\} }_{i = 1}^{n} \) and \( {\left\{ {x}_{j}\right\} }_{j = 1}^{m} \) of \( X \) satisfy the condition\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{n}{\left| {x}^{ * }\left( {z}_{i}\right) \right| }^{2} \leq \mathop{\sum }\lim... | Proof. Let \( F \) be the linear span of \( \left\{ {{x}_{1},\ldots ,{x}_{m},{z}_{1},\ldots ,{z}_{n}}\right\} \) in \( X \) . By hypothesis, the quadratic form \( Q \) defined on \( {F}^{ * } \) by\n\n\[ Q\left( {f}^{ * }\right) = \mathop{\sum }\limits_{{j = 1}}^{m}{\left| {f}^{ * }\left( {x}_{j}\right) \right| }^{2} -... | Yes |
Lemma 7.4.6. If \( X \) and \( Y \) are two isomorphic finite-dimensional Banach spaces, then\n\n\[ d\left( {X, Y}\right) = \min \left\{ {\parallel T\parallel \begin{Vmatrix}{T}^{-1}\end{Vmatrix} : T : X \rightarrow Y\text{ is an isomorphism }}\right\} . | Proof. Pick out \( C > d\left( {X, Y}\right) \) and consider\n\n\[ \mathcal{K} = \{ T \in \mathcal{B}\left( {X, Y}\right) : \parallel x\parallel \leq \parallel T\left( x\right) \parallel \leq C\parallel x\parallel ,\forall x \in X\} .\n\nThe set \( \mathcal{K} \) is a nonempty compact subset of \( \mathcal{B}\left( {X,... | Yes |
Theorem 7.4.8 (Maurey). Let \( X \) be a Banach space of type 2. Let \( Y \) be a closed subspace of \( X \) that is isomorphic to a Hilbert space. Then \( Y \) is complemented in \( X \) . | Proof. Since \( Y \) is of cotype 2, the identity map on \( Y \) can be extended to a projection of \( X \) onto \( Y \) . | Yes |
Problem 7.4.9. Suppose \( X \) is a Banach space with the property that for every closed subspace \( E \) of \( X \) and every operator \( {T}_{0} : E \rightarrow H \) ( \( H \) a Hilbert space) there is a bounded extension \( T : X \rightarrow H \) . Must \( X \) be a space of type 2 ? | For a partial positive solution of this problem we refer to [44]. | No |
Theorem 8.1.3. Let \( K \) and \( L \) be two compact Hausdorff spaces and let \( B : \mathcal{C}\left( K\right) \times \mathcal{C}\left( L\right) \rightarrow \mathbb{R} \) be a bounded bilinear form. Then for every \( {\left( {f}_{k}\right) }_{k = 1}^{n} \) in \( \mathcal{C}\left( K\right) \) and \( {\left( {g}_{k}\ri... | Proof. The proof relies on a partition of unity argument. Let \( {\left( {f}_{k}\right) }_{k = 1}^{n} \) a sequence in \( \mathcal{C}\left( K\right) \) and \( {\left( {g}_{k}\right) }_{k = 1}^{n} \) be a sequence in \( \mathcal{C}\left( L\right) \). Given \( \delta > 0 \), one can find a finite open cover \( {\left( {U... | Yes |
Theorem 8.1.5. Suppose \( K \) is a compact Hausdorff space, that \( \left( {\Omega ,\mu }\right) \) is a \( \sigma \) -finite measure space, and that \( T : \mathcal{C}\left( K\right) \rightarrow {L}_{1}\left( \mu \right) \) is a continuous operator. Then for every finite sequence \( {\left( {f}_{k}\right) }_{k = 1}^{... | Proof. Let us define a bilinear form \( B : \mathcal{C}\left( K\right) \times {L}_{\infty }\left( \mu \right) \rightarrow \mathbb{R} \) by\n\n\[ \nB\left( {f, g}\right) = {\int }_{\Omega }g \cdot T\left( f\right) {d\mu } \n\]\n\nGiven a sequence \( {\left( {f}_{k}\right) }_{k = 1}^{n} \) in \( \mathcal{C}\left( K\right... | Yes |
Theorem 8.1.6. Suppose \( K \) is a compact Hausdorff space, that \( \left( {\Omega ,\mu }\right) \) is a probability measure space, and that \( T : \mathcal{C}\left( K\right) \rightarrow {L}_{1}\left( \mu \right) \) is a continuous operator. Then there exists a density function \( h \) on \( \Omega \) such that for al... | Proof. It is enough to note that Theorem 8.1.5 implies that \[ {\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{i = 1}}^{n}{\left| T{f}_{i}\right| }^{2}\right) }^{1/2}\end{Vmatrix}}_{1} \leq {K}_{G}\parallel T\parallel {\left( \mathop{\sum }\limits_{{i = 1}}^{n}{\begin{Vmatrix}{f}_{i}\end{Vmatrix}}_{\infty }^{2}\right) ... | Yes |
Theorem 8.2.4. Let \( T \) be an operator between the Banach spaces \( X \) and \( Y \) . If \( 1 \leq \) \( r < p < \infty \) and \( T \) is \( r \) -absolutely summing, then \( T \) is p-absolutely summing with \( {\pi }_{p}\left( T\right) \leq {\pi }_{r}\left( T\right) \) | Proof. Given \( p > r \), let us pick \( q \) such that \( 1/p + 1/q = 1/r \) . Suppose \( {\left( {x}_{i}\right) }_{i = 1}^{n} \) in \( X \) satisfy\n\n\[ \n{\left( \mathop{\sum }\limits_{{i = 1}}^{n}{\left| {x}^{ * }\left( {x}_{i}\right) \right| }^{p}\right) }^{1/p} \leq 1,\;\forall {x}^{ * } \in {B}_{{X}^{ * }}.\n\]... | Yes |
Theorem 8.2.7. Let \( K \) be a compact Hausdorff space and let \( \mu \) be a \( \sigma \) -finite measure. Then every bounded operator \( T : \mathcal{C}\left( K\right) \rightarrow {L}_{1}\left( \mu \right) \) is 2-absolutely summing with \( {\pi }_{2}\left( T\right) \leq {K}_{G}\parallel T\parallel \) . | Proof. Using Lemma 6.2.16 in combination with Theorem 8.1.5, we obtain\n\n\[ \n{\left( \mathop{\sum }\limits_{{i = 1}}^{n}{\begin{Vmatrix}T{f}_{i}\end{Vmatrix}}_{1}^{2}\right) }^{1/2} = {\left( \mathop{\sum }\limits_{{i = 1}}^{n}{\begin{Vmatrix}{\left| T{f}_{i}\right| }^{2}\end{Vmatrix}}_{1/2}\right) }^{1/2} \n\]\n\n\[... | Yes |
Theorem 8.2.8. Suppose \( X \) is a closed subspace of \( \mathcal{C}\left( K\right) \) ( \( K \) compact Hausdorff). An operator \( T \) from \( X \) into a Banach space \( Y \) is p-absolutely summing for some \( 1 \leq p < \infty \) with \( {\pi }_{p}\left( T\right) \leq C \) if and only if there is a regular Borel ... | Proof. Assume first that \( 0 \neq T \) is a \( p \) -absolutely summing operator. We will use Lemma 7.3.5 to find a linear functional \( \mathcal{L} \) on \( \mathcal{C}\left( K\right) \) satisfying\n\n\[ \mathcal{L}\left( f\right) \leq \mathop{\max }\limits_{{s \in K}}f\left( s\right) ,\;\forall f \in \mathcal{C}\lef... | Yes |
Theorem 8.2.13. Suppose that \( X, Y \) are Banach spaces and that \( E \) is a closed subspace of \( X \) . Suppose the operator \( T : E \rightarrow Y \) is 2-absolutely summing. Then there exists a 2-absolutely summing operator \( \widetilde{T} : X \rightarrow Y \) such that \( {\left. \widetilde{T}\right| }_{E} = T... | Proof. We can factor the operator \( T : E \rightarrow Y \) using Remark 8.2.11: \n\nOn the other hand, the natural inclusion \( {j}_{2} : \mathcal{C}\left( {B}_{{E}^{ * }}\right) \rightarrow {L}_{2}\left( {{B}_{{E}^... | Yes |
Theorem 8.2.14 (Dvoretzky-Rogers Theorem). Let \( X \) be a Banach space such that every unconditionally convergent series in \( X \) is absolutely convergent. Then \( X \) is finite-dimensional. | Proof. By Proposition 8.2.2, our hypothesis is equivalent to saying that the identity operator \( {I}_{X} : X \rightarrow X \) is absolutely summing; hence it is also 2-absolutely summing by Theorem 8.2.4. Now by Theorem 8.2.12 we deduce that \( X \) is isomorphic to a Hilbert space. But we have already seen that every... | Yes |
An operator \( T : {H}_{1} \rightarrow {H}_{2} \) is Hilbert-Schmidt if and only if \( T \) is 2-absolutely summing. Furthermore, \( \parallel T{\parallel }_{HS} = {\pi }_{2}\left( T\right) \) . | Proof. Suppose first that \( T \) is 2-absolutely summing. If \( {\left( {e}_{j}\right) }_{j = 1}^{\infty } \) is an orthonormal basis of \( {H}_{1} \), then for each \( n \in N \) we have\n\n\[ \sup \left\{ {{\left( \mathop{\sum }\limits_{{j = 1}}^{n}{\left| \left\langle {e}_{j}, x\right\rangle \right| }^{2}\right) }^... | Yes |
Theorem 8.3.1. Suppose \( T : {L}_{1}\left( \mu \right) \rightarrow {\ell }_{2} \) is a bounded operator. Then \( T \) is absolutely summing and \( {\pi }_{1}\left( T\right) \leq {K}_{G}\parallel T\parallel \) . | Proof. Suppose \( {\left( {f}_{i}\right) }_{i = 1}^{n} \) in \( {L}_{1}\left( \mu \right) \) are such that\n\n\[ \sup \left\{ {\mathop{\sum }\limits_{{i = 1}}^{n}\left| {{\int }_{\Omega }{f}_{i}{gd\mu }}\right| : g \in {L}_{\infty }\left( \mu \right) ,\parallel g{\parallel }_{\infty } \leq 1}\right\} \leq 1. \]\n\nWe m... | Yes |
Theorem 8.3.5. If \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) is a normalized unconditional basis of a Hilbert space, then \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) is equivalent to the canonical basis of \( {\ell }_{2} \) . | Proof. Let \( {\mathrm{K}}_{\mathrm{u}} \) be the unconditional basis constant of \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) . The unconditionality of the basis and the generalized parallelogram law yield\n\n\[ \begin{Vmatrix}{\mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i}{u}_{i}}\end{Vmatrix} \leq {\mathrm{K}}_{\mat... | Yes |
Proposition 8.3.7. If \( 1 < p < \infty, p \neq 2 \), then \( {\ell }_{p} \) has at least two non-equivalent unconditional bases. | Proof. Let \( 1 < p < \infty, p \neq 2 \) . We saw in Proposition 6.4.2 that the operator \( P \) defined in \( {L}_{p} \) by\n\n\[ P\left( f\right) = \mathop{\sum }\limits_{{k = 1}}^{\infty }\left( {{\int }_{0}^{1}f\left( t\right) {r}_{k}\left( t\right) {dt}}\right) {r}_{k} \]\n\nis a projection onto \( {R}_{p} \), th... | Yes |
Lemma 9.1.3. Let \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be a normalized perfectly homogeneous basis of a Banach space \( X \) . Then \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is uniformly equivalent to all its normalized constant-coefficient block basic sequences. That is, there is a constant \( \mathr... | Proof. It suffices to prove such an inequality for the basic sequence \( {\left( {e}_{n}\right) }_{n = {n}_{0} + 1}^{\infty } \) for some \( {n}_{0} \) . If the lemma fails, we can inductively build constant-coefficient block basic sequences \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) and \( {\left( {v}_{n}\right)... | Yes |
Lemma 9.1.4. Suppose that \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is a normalized unconditional basis of a Banach space \( X \) . If \( \mathop{\sup }\limits_{N}\lambda \left( N\right) < \infty\), then \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is equivalent to the canonical basis of \( {c}_{0} \) . | Proof. For every \( N \) and scalars \( {\left( {a}_{n}\right) }_{n = 1}^{N} \) we have\n\n\[ \n\frac{1}{{\mathrm{\;K}}_{\mathrm{u}}}\mathop{\sup }\limits_{n}\left| {a}_{n}\right| \leq \begin{Vmatrix}{\mathop{\sum }\limits_{{n = 1}}^{N}{a}_{n}{e}_{n}}\end{Vmatrix} \leq {\mathrm{K}}_{\mathrm{u}}\mathop{\sup }\limits_{n}... | Yes |
Lemma 9.1.5. Let \( {\left( {e}_{i}\right) }_{i = 1}^{\infty } \) be a normalized perfectly homogeneous basis of a Banach space \( X \) . Then, if \( \dot{\mathrm{K}} \) is the constant given by Lemma 9.1.3, we have\n\n\[ \frac{1}{{\mathrm{\;K}}^{3}}\lambda \left( n\right) \lambda \left( m\right) \leq \lambda \left( {n... | Proof. Consider a family \( {\left( {f}_{j}\right) }_{j = 1}^{m} \) of \( m \) disjoint blocks of length \( n \) of the basis \( {\left( {e}_{i}\right) }_{i = 1}^{\infty } \),\n\n\[ {f}_{j} = \mathop{\sum }\limits_{{i = \left( {j - 1}\right) n + 1}}^{{jn}}{e}_{i},\;j = 1,\ldots, m. \]\n\nLet \( {c}_{j} = \begin{Vmatrix... | Yes |
Lemma 9.1.6. Let \( {\left( {s}_{n}\right) }_{n = 1}^{\infty } \) be a sequence of real numbers.\n\n(i) Suppose that \( {s}_{m + n} \leq {s}_{m} + {s}_{n} \) for all \( m, n \in \mathbb{N} \) . Then \( \mathop{\lim }\limits_{n}{s}_{n}/n \) exists (possibly equal to \( - \infty \) ) and\n\n\[ \mathop{\lim }\limits_{{n \... | Proof. (i) Fix \( n \in \mathbb{N} \) . Then, each \( m \in \mathbb{N} \) can be written as \( m = \ln + r \) for some \( 0 \leq l \) and \( 0 \leq r < n \) . The hypothesis implies that\n\n\[ {s}_{ln} \leq l{s}_{n},\;{s}_{{ln} + r} \leq l{s}_{n} + {s}_{r}. \]\n\nThus\n\n\[ \frac{{s}_{m}}{m} = \frac{{s}_{{ln} + r}}{{ln... | Yes |
Lemma 9.1.7. Let \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be a normalized perfectly homogeneous basis of a Banach space \( X \) . Then, either \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is equivalent to the canonical basis of \( {c}_{0} \) or there exist a constant \( C \) and \( 1 \leq p < \infty \) such... | Proof. If we plug \( m = {2}^{k} \) and \( n = {2}^{j} \) in equation (9.4), we obtain\n\n\[ \n\frac{1}{{\mathrm{\;K}}^{3}}\lambda \left( {2}^{k}\right) \lambda \left( {2}^{j}\right) \leq \lambda \left( {2}^{j + k}\right) \leq {\mathrm{K}}^{3}\lambda \left( {2}^{k}\right) \lambda \left( {2}^{j}\right) . \n\]\n\n(9.6)\n... | Yes |
Lemma 9.2.2. Suppose \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is a symmetric basis of a Banach space \( X \) . Then there exists a constant \( D \) such that\n\n\[ \n{D}^{-1}\begin{Vmatrix}{\mathop{\sum }\limits_{{i = 1}}^{N}{a}_{i}{e}_{{j}_{i}}}\end{Vmatrix} \leq \begin{Vmatrix}{\mathop{\sum }\limits_{{i = 1}}... | Proof. It is enough to prove the lemma for the basic sequence \( {\left( {e}_{n}\right) }_{n \geq {n}_{0}} \) for some \( {n}_{0} \) . If it is false, then for every \( {n}_{0} \) we can build a strictly increasing sequence of natural numbers \( {\left( {p}_{n}\right) }_{n = 0}^{\infty } \) with \( {p}_{0} = 0 \), natu... | Yes |
Lemma 9.4.1. Let \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be an unconditional basis of a Banach space \( X \) . Suppose that \( {\left( {u}_{k}\right) }_{k = 1}^{\infty } \) is a normalized block basic sequence of \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) such that the subspace \( \left\lbrack {u}_{k}\ri... | Proof. Suppose\n\n\[ {u}_{k} = \mathop{\sum }\limits_{{j \in {A}_{k}}}{a}_{j}{e}_{j} \]\n\nwhere \( {A}_{k} = \operatorname{supp}{u}_{k} \), and that \( P \) is a bounded projection onto \( \left\lbrack {u}_{k}\right\rbrack \) . For each \( k \) let \( {Q}_{k} \) be the projection onto \( {\left\lbrack {e}_{j}\right\rb... | Yes |
Theorem 9.4.4. Let \( X \) be a Banach space with unconditional basis. If every closed subspace of \( X \) is complemented in \( X \), then \( X \) is isomorphic to \( {\ell }_{2} \) . | Proof. Let \( {\left( {x}_{n}\right) }_{n = 1}^{\infty } \) be an unconditional basis of such an \( X \) . By Theorem 9.4.2, \( {\left( {x}_{n}\right) }_{n = 1}^{\infty } \) is equivalent either to the canonical basis of \( {c}_{0} \) or to the canonical basis of \( {\ell }_{p} \) for some \( 1 \leq p < \infty \) .\n\n... | Yes |
Lemma 9.5.4. Let \( E, F \) be two closed subspaces of codimension 1 of a Banach space \( X \) . Then there exists an isomorphism \( T : E \rightarrow F \) such that \( \parallel T\parallel \begin{Vmatrix}{T}^{-1}\end{Vmatrix} \leq {25} \) . | Proof. Unless \( E = F, E \cap F \) is a subspace of \( X \) of codimension 2. Let us pick \( {x}_{0} \in E \smallsetminus \left( {E \cap F}\right) \) such that \( 1 = \begin{Vmatrix}{x}_{0}\end{Vmatrix}d\left( {{x}_{0}, E \cap F}\right) \leq 2 \) . Analogously, pick \( {x}_{1} \in F \) such that \( 1 = \begin{Vmatrix}... | Yes |
Lemma 9.5.5. Suppose that \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is a basis of a Banach space \( X \) and that \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) is a block basic sequence of \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) . Then there exists a basis \( {\left( {f}_{n}\right) }_{n = 1}^{\infty ... | Proof. For each \( n \in \mathbb{N} \) suppose that \( {u}_{n} \) is normalized and supported on the basis elements \( \left\{ {{e}_{{r}_{n - 1} + 1},\ldots ,{e}_{{r}_{n}}}\right\} \), where \( {\left( {r}_{n}\right) }_{n = 1}^{\infty } \) is an increasing sequence of positive integers with \( {r}_{1} = 1 \) . Let \( {... | Yes |
Theorem 9.5.6 (Pelczyński-Singer). Let \( X \) be any Banach space with a basis. Then \( X \) has a conditional basis. | Proof. Assume that every basis of \( X \) is unconditional and let \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be one of them. Suppose \( {\left( {u}_{k}\right) }_{k = 1}^{\infty } \) is a block basic sequence of \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \). Then, using Lemma 9.5.5, \( X \) has a basis \( {\le... | Yes |
Theorem 10.2.3. A basis \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) in a Banach space \( X \) is quasi-greedy if and only if there is a constant \( \mathrm{C} \geq 1 \) such that \( \begin{Vmatrix}{{\mathcal{G}}_{m}\left( x\right) }\end{Vmatrix} \leq \mathrm{C}\parallel x\parallel \) for all \( x \in... | The proof of Theorem 10.2.3 relies on Lemmas 10.2.5 and 10.2.6. We will also use several times a simple argument that we record in Lemma 10.2.4. Given a finite subset \( A \subset \mathbb{N} \), we denote by \( {P}_{A} : X \rightarrow X \) the (bounded and linear) projection onto the vector space \( \left\lbrack {{e}_{... | Yes |
Lemma 10.2.5. Let \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be a basis in a Banach space \( X \) . The following are equivalent:\n\n(a) \( {G}_{m}\left( x\right) \rightarrow x \) for every \( x \in X \) and every greedy approximation \( {\left( {G}_{m}\left( x\right) \right) }_{m = 1}^{\infty } \ .... | Proof. The implications \( \left( a\right) \Rightarrow \left( b\right) \Rightarrow \left( c\right) \Rightarrow \left( d\right) \) are obvious, and \( \left( d\right) \Leftrightarrow \left( e\right) \) is an easy consequence of (10.1). To complete the chain of implications, let us prove (a) with the assumption of (d).\n... | Yes |
Lemma 10.2.6. Let \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be a basis in a Banach space \( X \) . The following are equivalent:\n\n(a) There exists a constant \( \mathrm{C} \) such that for all \( x \in X \), all \( m \in \mathbb{N} \), and all greedy sums \( {G}_{m}\left( x\right) \),\n\n\[ \begi... | Proof. The implications \( \left( a\right) \Rightarrow \left( b\right) \Rightarrow \left( c\right) \Rightarrow \left( d\right) \Rightarrow \left( e\right) \) (maintaining the constant \( \mathrm{C} \) ) are obvious. Let us show that \( \left( e\right) \Rightarrow \left( a\right) \) with the same constant. To that end, ... | Yes |
Lemma 10.2.7. Suppose that (e) in Lemma 10.2.6 does not hold. Then for every positive constant \( \mathrm{C} \) and for every finite set \( A \subset \mathbb{N} \), there exists \( x \in X \) with \( \left| {\operatorname{supp}\left( x\right) }\right| < \infty \) and \( \operatorname{supp}\left( x\right) \cap A = \varn... | Proof. Fix a constant \( \mathrm{C} > 0 \) . Given any finite \( A \subset \mathbb{N} \), put \( M = \mathop{\max }\limits_{{E \subset A}}\begin{Vmatrix}{P}_{E}\end{Vmatrix} \) . By our assumption there exists a finitely supported \( y \) in \( X \) and a strictly greedy sum of \( y,{G}_{r}\left( y\right) = {P}_{F}\lef... | Yes |
Proposition 10.2.10. Suppose that \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is a quasi-greedy basis. Then:\n\n(a) Whenever \( A \) and \( B \) are finite subsets of integers with \( B \subset A \), \n\n\[ \begin{Vmatrix}{\mathop{\sum }\limits_{{n \in B}}{e}_{n}}\end{Vmatrix} \leq {\mathrm{C}}_{\mat... | Proof. Let \( B \subset A \subset \mathbb{N} \), with \( A \) finite. Note that both \( g = \mathop{\sum }\limits_{{n \in B}}{e}_{n} \) and \( h = \) \( \mathop{\sum }\limits_{{n \in A \smallsetminus B}}{e}_{n} \) are greedy sums of \( x = \mathop{\sum }\limits_{{n \in A}}{e}_{n} \) . In the same way, both \( g \) and ... | Yes |
Corollary 10.2.11. Suppose \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is quasi-greedy. For every \( A \subset \mathbb{N} \) finite and real numbers \( {\left( {a}_{n}\right) }_{n \in A} \) , \[ \begin{Vmatrix}{\mathop{\sum }\limits_{{n \in A}}{a}_{n}{e}_{n}}\end{Vmatrix} \leq 2{\mathbf{C}}_{\mathrm{... | Proof. The result follows from (10.6) by convexity. | No |
Theorem 10.2.12. Let \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be a quasi-greedy basis in a Banach space \( X \). (a) For every \( x \in X \), every greedy ordering \( \pi \) of \( x \), and every \( m \in \mathbb{N} \), \[ \left| {{e}_{\pi \left( m\right) }^{ * }\left( x\right) }\right| \begin{Vma... | Proof. (a) For \( 1 \leq j \leq m \), let \( {G}_{j}\left( x\right) = \mathop{\sum }\limits_{{k = 1}}^{j}{e}_{\pi \left( k\right) }^{ * }\left( x\right) {e}_{\pi \left( k\right) } \) be a greedy sum of order \( j \) of \( x \). Put \( {b}_{j} = 1/\left| {{e}_{\pi \left( j\right) }^{ * }\left( x\right) }\right| \) and p... | Yes |
Corollary 10.2.13. Suppose \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is a quasi-greedy basis of \( X \) with quasi-greedy constant \( {\mathrm{C}}_{\mathrm{{qg}}} \) . For every \( x \in X \) and every finite subset \( A \subset \operatorname{supp}\left( x\right) \) we have\n\n\[ \begin{Vmatrix}{{P... | Proof. Let \( \mu = \min \left\{ {\left| {{e}_{n}^{ * }\left( x\right) }\right| : n \in A}\right\} \) and \( \nu = \max \left\{ {\left| {{e}_{n}^{ * }\left( x\right) }\right| : n \in A}\right\} \) and consider \( B = \left\{ {n \in \mathbb{N} : \mu \leq \left| {{e}_{n}^{ * }\left( x\right) }\right| \leq v}\right\} \) .... | Yes |
Theorem 10.2.14. If \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is quasi-greedy, then\n\n\[{\mathrm{k}}_{m} = \mathcal{O}\left( {{\log }_{2}\left( m\right) }\right)\] | Proof. Consider an integer \( m \geq 2 \) and let \( p = \left\lfloor {{\log }_{2}\left( m\right) }\right\rfloor \), be such that \( {2}^{p} \leq m < \) \( {2}^{p + 1} \) . Let \( x \in X \) with \( \parallel x\parallel = 1 \) so that \( \left| {{e}_{n}^{ * }\left( x\right) }\right| \leq \mathrm{K} \) for all \( n \in ... | Yes |
Proposition 10.2.16. Let \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be a quasi-greedy basis in a Banach space \( X \) . Suppose \( {\left( {\lambda }_{n}\right) }_{n = 1}^{\infty } \) is a sequence of real numbers such that \( 0 < \mathop{\inf }\limits_{n}\left| {\lambda }_{n}\right| \leq \mathop{\sup }\limits_{n... | Proof. Let \( a = \mathop{\sup }\limits_{n}\left| {\lambda }_{n}\right| \) and assume (by homogeneity) that \( \mathop{\inf }\limits_{n}\left| {\lambda }_{n}\right| = 1 \) . Let \( {G}_{m}\left\lbrack {\widetilde{\mathcal{B}}, X}\right\rbrack \left( x\right) = {P}_{A}\left( x\right) \) be a greedy sum of \( x \) with r... | Yes |
Lemma 10.3.4. Let \( \\mathcal{B} \) be a basis with basis constant \( {\\mathrm{K}}_{\\mathrm{b}} = \\mathop{\\sup }\\limits_{{m \\in \\mathbb{N}}}\\begin{Vmatrix}{S}_{m}\\end{Vmatrix} \) . Then:\n\n(a) The functions \( {\\varphi }_{u} \) and \( {\\varphi }_{l} \) are essentially nondecreasing, i.e., for \( m \\leq r ... | Proof. Let us see (b) and leave the proof of (a) as an exercise. For every finite set \( A \) with \( \\left| A\\right| = m \\geq 2 \) let us write\n\n\[ \n\\mathop{\\sum }\\limits_{{n \\in A}}{e}_{n} = \\frac{1}{m - 1}\\mathop{\\sum }\\limits_{{k \\in A}}\\mathop{\\sum }\\limits_{{n \\in A\\smallsetminus \\{ k\\} }}{e... | No |
Lemma 10.3.6. Let \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) and \( \widetilde{\mathcal{B}} = {\left( {\widetilde{e}}_{n}\right) }_{n = 1}^{\infty } \) be quasi-greedy bases for the Banach spaces \( X \) and \( Y \) respectively. Let \( A \) and \( B \) be finite sets of integers. Suppose that \( x ... | Proof. The result follows readily from (10.7) and (10.9). | Yes |
Proposition 10.3.8. Suppose that \( \mathcal{B} \) is a quasi-greedy basis in a Banach space \( X \) . If \( {\left( {\varphi }_{u}\left( m\right) \right) }_{m = 1}^{\infty } \) is bounded then \( \mathcal{B} \) is equivalent to the canonical \( {c}_{0} \) basis. | Proof. By Corollary 10.2.11, the proof is analogous to the proof of Lemma 9.1.4. | No |
Consider \( X = {\ell }_{p} \oplus {\ell }_{q},1 \leq p < q < \infty \), and let \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be the direct sum of the natural unit vector bases of the two spaces. That is, in our basis we have\n\n\[ \begin{Vmatrix}{\mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{e}_{n... | Fix a small \( \delta > 0 \) and for each \( m \) define a vector \( {z}_{m} = \mathop{\sum }\limits_{{k = 1}}^{{2m}}\left( {1 + {\left( -1\right) }^{k}\delta }\right) {e}_{k} \) . We have \( {\mathcal{G}}_{m}\left( {z}_{m}\right) = \mathop{\sum }\limits_{{k = 1}}^{m}\left( {1 + \delta }\right) {e}_{2k} \), whence \( \... | Yes |
Corollary 10.4.6. Every subsymmetric basis in a Banach space is greedy. | Proof. Suppose that \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is subsymmetric and nongreedy. Since \( \mathcal{B} \) is unconditional, it must fail to be democratic. We recursively construct sequences of mutually disjoint subsets of integers \( {\left( {A}_{k}\right) }_{k = 1}^{\infty } \) and \( {... | Yes |
Proposition 10.4.7. The normalized Haar system \( {\mathcal{H}}_{p} = {\left( {h}_{n}^{p}\right) }_{n = 1}^{\infty } \) is a greedy basis in \( {L}_{p}\left\lbrack {0,1}\right\rbrack \) for \( 1 < p < \infty \) . | Proof. Since \( {\mathcal{H}}_{p} \) is unconditional (Theorem 6.1.7), by Theorem 10.4.5 we need only show that \( {\mathcal{H}}_{p} \) is democratic. To this end, by Lemma 10.3.5 it will be enough to estimate \( \begin{Vmatrix}{\mathop{\sum }\limits_{{n \in A}}{h}_{n}^{p}}\end{Vmatrix} \) for finite subsets of integer... | No |
Lemma 10.4.8. Let \( 1 < r < \infty \) and \( 0 < p < \infty \) . There are positive constants \( {\mathbf{c}}_{r, p} \) and \( {\mathbf{C}}_{r, p} \) such that for every finite set of integers \( A \) , | \[ {\mathrm{C}}_{r, p}{\left( \mathop{\sum }\limits_{{k \in A}}{r}^{pk}\right) }^{1/p} \leq {\left( \mathop{\sum }\limits_{{k \in A}}{r}^{2k}\right) }^{1/2} \leq {\mathrm{C}}_{r, p}{\left( \mathop{\sum }\limits_{{k \in A}}{r}^{pk}\right) }^{1/p}. \] | No |
Theorem 10.5.3. A basis \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) in a Banach space \( X \) is almost greedy if and only if it is quasi-greedy and democratic. | Proof. We have already seen that almost greedy bases are quasi-greedy. The proof that greedy bases are democratic (see Theorem 10.4.5) carries over to show that almost greedy bases are democratic by replacing \( {\mathrm{C}}_{\mathrm{g}} \) with \( {\mathrm{C}}_{\mathrm{{ag}}} \) .\n\nFor the converse, let \( x \in X \... | Yes |
An almost greedy basis that is not greedy. | Aside from being quasi-greedy, the basis \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) in Example 10.2.9 is democratic. Indeed, if \( \left| A\right| = m \), then\n\n\[ \n{\left( \mathop{\sum }\limits_{{n \in A}}1\right) }^{1/2} = {m}^{1/2} \n\] \n\nwhile \n\n\[ \n\mathop{\sum }\limits_{{n \in A}}\frac... | Yes |
Theorem 10.5.5. Suppose \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is a basis in a Banach space \( X \) . The following conditions are equivalent:\n\n(a) \( \mathcal{B} \) is almost greedy.\n\n(b) For every \( \lambda > 1 \) there exists a constant \( {\mathrm{C}}_{\lambda } \) such that\n\n\[ \begi... | Proof. To show \( \left( a\right) \Rightarrow \left( b\right) \) we need a lemma that roughly speaking tells us that the gap between \( \widetilde{\sigma } \) and \( \sigma \) depends on the proximity between the democracy functions of the basis.\n\nLemma 10.5.6. Suppose | No |
Lemma 10.5.6. Suppose \( \mathcal{B} \) is quasi-greedy. Then, for all \( m, r \in \mathbb{N} \), \[ {\widetilde{\sigma }}_{m + r}\left( x\right) \leq \left( {1 + {\mathrm{C}}_{\mathrm{{qg}}} + {16}{\mathrm{C}}_{\mathrm{{qg}}}^{5}\frac{{\varphi }_{u}\left( m\right) }{{\varphi }_{l}\left( r\right) }}\right) {\sigma }_{m... | Proof. Take \( y \in {\sum }_{m} \) and let \( A = \operatorname{supp}\left( y\right) \) . Consider \( z = x - y \) . Pick \( {G}_{r}\left( z\right) = {P}_{B}\left( z\right) \) a greedy sum of \( z \) of order \( r \) . Since \( \left| A\right| \leq m \) and \( \left| B\right| = r \), there is a set \( E \subset \mathb... | Yes |
Theorem 10.6.3. Let \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be a greedy basis in a Banach space \( X \) . Suppose \( {\varphi }_{u}\left( m\right) \approx {m}^{\alpha } \) for some \( 0 < \alpha < 1 \) . Then \( {\mathcal{B}}^{ * } \) is also greedy. | Proof. From Theorem 10.4.5 we know that \( \mathcal{B} \) is unconditional, say with unconditional constant \( {\mathrm{K}}_{\mathrm{u}} \) ; hence the basic sequence \( {\mathcal{B}}^{ * } \) is also unconditional. To show that \( {\mathcal{B}}^{ * } \) is democratic under our assumption on \( {\varphi }_{u} \), by Th... | Yes |
Corollary 10.6.4. If \( \mathcal{B} \) is a greedy basis in \( {L}_{p}\left\lbrack {0,1}\right\rbrack ,1 < p < \infty \), then \( {\mathcal{B}}^{ * } \) is a greedy basis in \( {L}_{q}\left\lbrack {0,1}\right\rbrack \), where \( 1/p + 1/q = 1 \) . | Proof. Fix \( 1 < p < \infty \) . Notice that in \( {L}_{p}\left\lbrack {0,1}\right\rbrack \), all greedy bases \( \mathcal{B} \) have essentially the same democracy functions, namely\n\n\[ \n{\varphi }_{l}\left\lbrack {\mathcal{B},{L}_{p}}\right\rbrack \left( m\right) \approx {\varphi }_{u}\left\lbrack {\mathcal{B},{L... | Yes |
Example 10.6.5. A greedy basis \( \mathcal{B} \) such that \( {\mathcal{B}}^{ * } \) is not democratic (hence non-greedy). | Let \( {\mathcal{H}}_{1} = {\left( {h}_{n}^{1}\right) }_{n = 1}^{\infty } \) be the Haar system normalized in \( {L}_{1}\left\lbrack {0,1}\right\rbrack \) . Since \( {\mathcal{H}}_{1} \) is not unconditional, we cannot count on having an estimate like (10.16) for \( p = 1 \) . However, if we consider the space \( X \) ... | Yes |
(i) Suppose \( r \in \mathbb{N} \) and \( f : {\mathcal{F}}_{r}\left( \mathbb{N}\right) \rightarrow \mathbb{R} \) is a bounded function. Then there exists \( M \in {\mathcal{P}}_{\infty }\left( \mathbb{N}\right) \) such that \( \mathop{\lim }\limits_{{A \in {\mathcal{F}}_{r}\left( M\right) }}f\left( A\right) \) exists. | The proof of \( \left( i\right) \) is done by induction on \( r \) . For \( r = 1 \) it is trivially true. Assume that \( r \geq 2 \) and that (i) holds for \( r - 1 \) ; we must deduce that (i) is also true for \( r \) .\n\nFor distinct integers \( {m}_{1},\ldots ,{m}_{r} \), put\n\n\[ f\left( {{m}_{1},{m}_{2},\ldots ... | Yes |
Theorem 11.3.1 (James’s \( {\ell }_{1} \) distortion theorem). Let \( {\left( {x}_{n}\right) }_{n = 1}^{\infty } \) be a normalized basic sequence in a Banach space \( X \) that is equivalent to the canonical \( {\ell }_{1} \) -basis. Then given \( \epsilon > 0 \), there is a normalized block basic sequence \( {\left( ... | Proof. For each \( n \) let \( {M}_{n} \) be the least constant such that if \( {\left( {a}_{k}\right) }_{k = 1}^{\infty } \in {c}_{00} \) with \( {a}_{k} = 0 \) for \( k \leq n \), then\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }\left| {a}_{k}\right| \leq {M}_{n}\begin{Vmatrix}{\mathop{\sum }\limits_{{k = 1}}^{\in... | Yes |
Every Banach space \( X \) (not necessarily separable) is finitely representable in \( {c}_{0} \) . | Indeed, given any finite-dimensional subspace \( E \) of \( X \) and \( \epsilon > 0 \) , pick \( v \) such that \( \frac{1}{1 - v} < 1 + \epsilon \) and \( \left\{ {{e}_{1}^{ * },\ldots ,{e}_{N}^{ * }}\right\} \) a \( v \) -net in \( {B}_{{E}^{ * }} \) . Consider the mapping \( T : E \rightarrow {\ell }_{\infty }^{N} ... | No |
Proposition 12.1.4. If \( X \) is finitely representable in \( Y \), and \( Y \) is finitely representable in \( Z \), then \( X \) is finitely representable in \( Z \). | Proof. Suppose \( E \) is a finite-dimensional subspace of \( X \) and \( \epsilon > 0 \). Then there exists a finite-dimensional subspace \( F \) of \( Y \) with \( d\left( {E, F}\right) < {\left( 1 + \epsilon \right) }^{1/2} \). Similarly, we can find a finite-dimensional subspace \( G \) of \( Z \) such that \( d\le... | Yes |
Theorem 12.1.6. Given a (not necessarily separable) Banach space \( X \), the following are equivalent:\n\n(i) \( X \) is crudely finitely representable in \( {\ell }_{2} \).\n\n(ii) There exists a constant \( \lambda \) such that \( {d}_{E} \leq \lambda \) for every finite-dimensional subspace \( E \subset X \).\n\n(i... | Proof. (iii) \( \Rightarrow \left( i\right) \) is obvious. \( \left( i\right) \Rightarrow \left( {ii}\right) \) follows from the (also obvious) fact that every \( n \) -dimensional subspace of \( {\ell }_{2} \) is isometric to \( {\ell }_{2}^{n} \) . To obtain (iii) under the assumption of \( \left( {ii}\right) \), it ... | Yes |
Lemma 12.1.7. Suppose \( X \) is a separable Banach space and that \( {\left( {E}_{n}\right) }_{n = 1}^{\infty } \) is an increasing sequence of subspaces of \( X \) such that \( { \cup }_{n = 1}^{\infty }{E}_{n} \) is dense in \( X \) .\n\n(ii) Let \( \lambda > 1 \) and suppose that \( X \) has the property that given... | Proof. It is enough to prove (ii). Suppose \( X \) satisfies the property in the hypothesis that \( E \) is a finite-dimensional subspace of \( X \) and that \( {\left( {e}_{j}\right) }_{j = 1}^{N} \) is a basis of \( E \) with basis constant \( {\mathrm{K}}_{\mathrm{b}} \) . Then we can find \( n \) such that there ex... | Yes |
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