Math Problems Vault

Problem 6.1

Prove Lemma 6.1.

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Remark.

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Lemma 6.1: Given a sequence $\left\{ x_{n}\right\}$ in a Banach space $X$ , the following two statements are equivalent: (a) $\left\{ x_{n}\right\}$ is an unconditional basis for $X$ ; (b) $\left\{ x_{\sigma (n)}\right\}$ is a basis for $X$ for every permutation $\sigma$ of $\mathbb {N}$ . In this case, if $\left\{ a_{n}\right\}$ is the sequence of coefficient functionals for $\left\{ x_{n}\right\}$ , then $\left\{ a_{\sigma (n)}\right\}$ is the sequence of coefficient functionals for $\left\{ x_{\sigma (n)}\right\}$ .

Problem 6.2

(a)(b)

Prove Lemma 6.2: let $T: X \rightarrow Y$ be a topological isomorphism of Banach spaces and let $\left\{ x_{n}\right\}$ be a basis for $X$ .

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(a)

If $\left\{ x_{n}\right\}$ is an unconditional basis for $X$ , then $\left\{ T x_{n}\right\}$ is an unconditional basis for $Y$ .

(b)

If $\left\{ x_{n}\right\}$ is a bounded unconditional basis for $X$ , then $\left\{ T x_{n}\right\}$ is a bounded unconditional basis for $Y$ .

Problem 6.3

Prove Theorem 6.4.

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Remark.

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Notation 6.3: Let $\left\{ x_{n}\right\}$ be an unconditional basis for a Banach space $X$ , and let $\left\{ a_{n}\right\}$ be the biorthogonal system to $\left\{ x_{n}\right\}$ . To each finite set $F \subseteq \mathbb {N}$ we associate the partial sum operator $S_{F}(x) = \sum_{n \in F} \left\langle x, a_{n} \right\rangle x_{n}$ ; to each finite set $F \subseteq \mathbb {N}$ and each set of scalars $\mathcal{E} = \left\{ \varepsilon_{n}\right\}_{n \in F}$ satisfying $\varepsilon_{n} = \pm 1$ for all $n$ , the operator $S_{F, \mathcal{E}}(x) = \sum_{n \in F} \varepsilon_{n} \left\langle x, a_{n} \right\rangle x_{n}$ ; and to each finite set $F \subseteq \mathbb {N}$ and each collection of scalars $\Lambda = \left\{ \lambda_{n}\right\}_{n \in F}$ satisfying $\left|\lambda_{n}\right| \leq 1$ for all $n$ , the operator $S_{F, \Lambda }(x) = \sum_{n \in F} \lambda_{n} \left\langle x, a_{n} \right\rangle x_{n}$ (all for $x \in X$ ). The suprema below are taken over all such $F$ , $\mathcal{E}$ , $\Lambda$ .

Theorem 6.4: If $\left\{ x_{n}\right\}$ is an unconditional basis for a Banach space $X$ , then the following statements hold.

The following three quantities are finite for each $x \in X$ :
$|||x||| = \sup _{F}\left\| S_{F}(x)\right\| , \quad |||x|||_{\mathcal{E}} = \sup _{F, \mathcal{E}}\left\| S_{F, \mathcal{E}}(x)\right\| , \quad |||x|||_{\Lambda } = \sup _{F, \Lambda }\left\| S_{F, \Lambda }(x)\right\| .$
The following three numbers are finite:
$\mathcal{K} = \sup _{F}\left\| S_{F}\right\| , \quad \mathcal{K}_{\mathcal{E}} = \sup _{F, \mathcal{E}}\left\| S_{F, \mathcal{E}}\right\| , \quad \mathcal{K}_{\Lambda } = \sup _{F, \Lambda }\left\| S_{F, \Lambda }\right\| .$
$|||\cdot ||| \leq |||\cdot |||_{\mathcal{E}} \leq 2\, |||\cdot |||$ and $\mathcal{K} \leq \mathcal{K}_{\mathcal{E}} \leq 2 \mathcal{K}$ .
If $\mathbf{F} = \mathbb {R}$ then $|||\cdot |||_{\mathcal{E}} = |||\cdot |||_{\Lambda }$ and $\mathcal{K}_{\mathcal{E}} = \mathcal{K}_{\Lambda }$ .
If $\mathbf{F} = \mathbb {C}$ then $|||\cdot |||_{\mathcal{E}} \leq |||\cdot |||_{\Lambda } \leq 2\, |||\cdot |||_{\mathcal{E}}$ and $\mathcal{K}_{\mathcal{E}} \leq \mathcal{K}_{\Lambda } \leq 2 \mathcal{K}_{\mathcal{E}}$ .
$|||\cdot |||$ , $|||\cdot |||_{\mathcal{E}}$ , and $|||\cdot |||_{\Lambda }$ form norms on $X$ , each equivalent to the initial norm $\left\| \cdot \right\|$ , with
$\left\| \cdot \right\| \leq |||\cdot ||| \leq \mathcal{K}\left\| \cdot \right\| , \quad \left\| \cdot \right\| \leq |||\cdot |||_{\mathcal{E}} \leq \mathcal{K}_{\mathcal{E}}\left\| \cdot \right\| , \quad \left\| \cdot \right\| \leq |||\cdot |||_{\Lambda } \leq \mathcal{K}_{\Lambda }\left\| \cdot \right\| .$

Problem 6.4

(a)(b)

Let $\left\{ x_{n}\right\}$ be an unconditional basis for a Banach space $X$ , with associated coefficient functionals $\left\{ a_{n}\right\}$ .

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(a)

Prove that $\left\{ a_{n}\right\}$ is an unconditional basic sequence in $X^{*}$ .

(b)

Show that if $X$ is reflexive, then $\left\{ a_{n}\right\}$ is an unconditional basis for $X^{*}$ .

Problem 6.5

(a)(b)

(See Chapter 14 for proof that $\left\{ e^{2 \pi i n t}\right\}_{n \in \mathbb {Z}}$ is a basis for $L^{p}(\mathbf{T})$ when $1 < p < \infty$ , but is not a basis for $L^{1}(\mathbf{T})$ or $C(\mathbf{T})$ .)

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(a)

Use Orlicz's Theorem to prove that $\left\{ e^{2 \pi i n t}\right\}_{n \in \mathbb {Z}}$ cannot be an unconditional basis for $L^{p}(\mathbf{T})$ when $1 \leq p < 2$ .

(b)

Argue by duality to show that it also cannot be an unconditional basis when $2 < p < \infty$ .

Problem 6.6

Let $X$ be a real Banach space, and suppose that $\left\{ x_{n}\right\}$ is an unconditional basis for $X$ with unconditional basis constant $\mathcal{K}_{\mathcal{E}} = 1$ . Given $x = \sum a_{n} x_{n}$ and $y = \sum b_{n} y_{n}$ in $X$ , declare that $x \leq y$ if $a_{n} \leq b_{n}$ for every $n$ . Show that $\leq$ is a partial order on $X$ , and $X$ is a Banach lattice in the sense of Definition 3.35. Using the notation of that definition, show that $x \vee y = \sum \max \left\{ a_{n}, b_{n}\right\} x_{n}$ , $x \wedge y = \sum \min \left\{ a_{n}, b_{n}\right\} x_{n}$ , and $\left|x\right| = \sum \left|a_{n}\right| x_{n}$ .

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Problem 6.7

Set $F = \mathbb {R}$ . The Haar system is an orthonormal basis for $L^{2}[0,1]$ , so by Exercise 6.6 there is a partial ordering $\leq$ on $L^{2}[0,1]$ induced by this unconditional basis. There is also the ordinary partial ordering $\leq$ on $L^{2}[0,1]$ defined by $f \leq g$ if $f(t) \leq g(t)$ for a.e. $t$ . Do these two orderings coincide?

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Problem 6.8

(a)(b)

Let $W$ be the hat function of height 1 supported on $[0,1]$ , and let $s_{n, k}(t) = W\left(2^{n} t-k\right)$ be the dilated and translated hat functions of the Schauder system in $C[0,1]$ . Select the subsequence

t_{1} = s_{0,0}, \quad t_{2} = s_{1,0}, \quad t_{3} = s_{2,1}, \quad t_{4} = s_{3,2}, \quad t_{5} = s_{4,5}, \quad t_{6} = s_{5,10}, \quad \ldots ,

the hat functions on the intervals $I_{1} = [0,1]$ , $I_{2} = \left[0, \frac{1}{2}\right]$ , $I_{3} = \left[\frac{1}{4}, \frac{1}{2}\right]$ , $I_{4} = \left[\frac{1}{4}, \frac{3}{8}\right]$ , $I_{5} = \left[\frac{5}{16}, \frac{3}{8}\right]$ , $I_{6} = \left[\frac{5}{16}, \frac{11}{32}\right]$ , etc., where we alternate choosing the left or right half of $I_{N-1}$ as the interval $I_{N}$ on which the hat function $t_{N}$ is supported. Let

g_{N} = \sum _{n = 1}^{N} t_{n}, \quad h_{N} = \sum _{n = 1}^{N}(-1)^{n+1} t_{n},

let $\mu_{N}$ be the midpoint of $I_{N}$ , let $a_{N} = g_{N}\left(\mu_{N}\right)$ be the global maximum of $g_{N}$ , and let $b_{N} = \left\| h_{N}\right\|_{\infty }$ . (For $N \geq 3$ one endpoint of $I_{N}$ is $\mu_{N-2}$ and the other is $\mu_{N-1}$ , and $a_{N} = 1+\frac{a_{N-1}+a_{N-2}}{2}$ .)

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(a)

Show that $a_{N} \rightarrow \infty$ .

(b)

Show that $0 \leq b_{N} \leq 2$ for each $N$ .