Characterization of weakly unconditionally Cauchy series

Question

I am reading from Bases in Banach spaces I by I. Singer. The below defintions and results are from this book.

Let us first recall that a series $\displaystyle \sum_{i=1}^\infty x_i$ in a Banach space $E$ is called $\sigma(E, E^*)$- unconditionally Cauchy or weakly unconditionally Cauchy, if $\displaystyle \sum_{i=1}^\infty |f(x_i)| < \infty$ for all $f \in E^*.$ A series $\sum_{i=1}^\infty f_i$ in a conjugate Banach space $E^*$ is called $\sigma(E^*, E)$-unconditionally Cauchy if $\displaystyle \sum_{i=1}^\infty |f_i(x)| < \infty$ for all $x \in E.$

Lemma For a series $\displaystyle \sum_{i=1}^\infty f_i$ in a conjugate Banach space $E^*$ the following statements are equivalent:

(1.) $\displaystyle \sum_{i=1}^\infty f_i$ is $\sigma(E^*, E)$-unconditionally Cauchy.

(2.) There exists a constant $C > 0$ such that $$\sum_{i=1}^\infty |f_i(x)| \leq C \|x\| \quad (x \in E).$$

(3.) There exists a constant $C > 0$ such that $$\left\Vert\sum_{i=1}^n \alpha_i f_i \right\Vert \leq C \quad (\alpha_i = 0 \text{ or } 1, i = 1, ..., n; n= 1,2,...).$$

(4.) There exists a constant $C > 0$ such that $$\left\Vert \sum_{i=1}^n \epsilon_i f_i \right\Vert \leq C \quad (\epsilon_i = \pm{1}, i = 1, ..., n; n= 1,2,...).$$

(5.) There exists a constant $C > 0$ such that \begin{eqnarray} \left \Vert \sum_{i=1}^n \beta_i f_i \right \Vert \leq C \quad (|\beta_i| \leq 1, i = 1, ..., n; n= 1,2,...). \end{eqnarray}

(6.) $\displaystyle \sum_{i=1}^\infty f_i$ is $\sigma(E^*, E^{**})$-unconditionally Cauchy.

Proof The first implication follows applying the uniform boundedness principle to the sequence of continuous non-linear functionals $\{p_n\},$ where $$p_n(x) = \displaystyle \sum_{i=1}^n |f_i(x)| (x \in E, n = 1, 2, \ldots ).$$ ... The implication $6 \rightarrow 1$ is obvious.

Corollary For a series $\displaystyle \sum_{i=1}^\infty x_i$ in a Banach space $E$ the following statements are equivalent:

(1.) $\displaystyle \sum_{i=1}^\infty x_i$ is weakly unconditionally Cauchy.

(2.) There exists a constant $C > 0$ such that \begin{eqnarray} \nonumber \displaystyle \sum_{i=1}^\infty |f(x_i)| \leq C \|f\| \quad (f \in E^*). \end{eqnarray}

(3.) There exists a constant $C > 0$ such that \begin{eqnarray} \nonumber \left \Vert \sum_{i=1}^n \alpha_i x_i \right \Vert \leq C \quad (\alpha_i = 0 \text{ or } 1, i = 1, ..., n; n= 1,2,...). \end{eqnarray}

(4.) There exists a constant $C > 0$ such that \begin{eqnarray} \nonumber \left \Vert \sum_{i=1}^n \epsilon_i x_i \right \Vert \leq C \quad (\epsilon_i = \pm{1}, i = 1, ..., n; n= 1,2,...). \end{eqnarray}

(5.) There exists a constant $C > 0$ such that \begin{eqnarray} \nonumber \left \Vert \sum_{i=1}^n \beta_i x_i \right \Vert \leq C \quad (|\beta_i| \leq 1 , i = 1, ..., n; n= 1,2,...). \end{eqnarray}

Proof This follows by embedding $E$ in $E^{**}$ and applying the previous lemma in $E^{**}.$

1. In the proof of the lemma, how did he apply the uniform boundedness principle to sequence of continuous non-linear functionals?

2. In the proof of the lemma, to prove $(6) \rightarrow (1)$ can we proceed as the following? For $x \in E.$ Let $C : E \rightarrow E^{**}$ denote the mapping that to each $x \in E$ assigns a linear function $g$ on $E^*$ such that $g(f) = f(x)$ for all $f \in E^*.$ So, $\sum_{i=1}^\infty |f_i(x)| = \sum_{i=1}^\infty |g(f_i)| < \infty.$

3. How does the proof of the corollary follow from the lemma via the embedding?

Any help will be appreciated. Thanks in advance.

Answer 1

For the first query, I assume the "first implication" refers to the implication $(1) \Rightarrow (2)$ in the lemma. There are other formulations of the uniform boundedness principle for nonlinear functions but I don't know which exact one the author intended to use.

Here is a direct argument. Let $E$ be a Banach space. Suppose that $(f_{i})_{i\in\mathbb{N}}$ is a sequence in $E^{*}$ and for each $x\in E$ the series $(\sum_{i=1}^{n} |f_{i}(x)|)_{n\in\mathbb{N}}$ converges. For each $n\in\mathbb{N}$ define $p_{n}\colon E\to [0, \infty )$ by $p_{n}(x) := \sum_{i=1}^{n} |f_{i}(x)|$. Also define \begin{equation} D := \{x\in E : p_{n}(x) \leq 1 \text{ for all } n\in \mathbb{N}\} . \end{equation} It is straightforward to check that $D$ is closed, convex and absorbing. Recall that every closed, convex, absorbing set in a Banach space is a neighbourhood of $0$. (See here or Theorem 1.3.14 of An Introduction to Banach Space Theory by Megginson.) It follows that there is some $\delta > 0$ such that if $x\in E$ and $\|x\| \leq \delta$ then $x\in D$ and hence $p_{n}(x) = \sum_{i=1}^{n} |f_{i}(x)| \leq 1$ for all $n\in\mathbb{N}$. From this it follows that \begin{equation} \sum_{i=1}^{\infty} |f_{i}(x)| \leq \delta^{-1} \|x\| \end{equation} for all $x\in E$.

For the second query, that is fine. Another way to say this is that if $J\colon E\to E^{**}$ is the canonical embedding given by $(Jx)(f) = f(x)$ then each of the following statements clearly imply the next one.

1. For each $F\in E^{**}$ the series $(\sum_{i=1}^{n} |F(f_{i})|)_{n\in\mathbb{N}}$ is convergent.

2. For each $x\in E$ the series $(\sum_{i=1}^{n} |(Jx)(f_{i})|)_{n\in\mathbb{N}}$ is convergent (because $J(E) \subseteq E^{**}$).

3. For each $x\in E$ the series $(\sum_{i=1}^{n} |f_{i}(x)|)_{n\in\mathbb{N}}$ is convergent (by the definition of $J$).

For the third query, we will just look at showing why two particular conditions are equivalent and it should be clear how the rest are equivalent from there. Let's say we want to show $(1) \Leftrightarrow (3)$ in the corollary. Let $E$ be a Banach space and $(x_{i})_{i\in\mathbb{N}}$ a sequence in $E$. Using that the canonical embedding $J\colon E\to E^{**}$ is an isometry it is clear that each statement below is equivalent to the next one.

1. The series $(\sum_{i=1}^{n} |f(x_{i})|)_{n\in\mathbb{N}}$ converges for every $f\in E^{*}$.

2. The series $(\sum_{i=1}^{n} |(Jx_{i})(f)|)_{n\in\mathbb{N}}$ converges for every $f\in E^{*}$. (Applying the definition of $J$.)

3. There exists a constant $C > 0$ such that $\| \sum_{i=1}^{n} \alpha_{i} Jx_{i} \|_{E^{**}} \leq C$ for all finite sequences $(\alpha_{i})_{i\in\mathbb{N}}$ in $\{0,1\}$ and $n\in\mathbb{N}$. (Applying $(1) \Leftrightarrow (3)$ from the lemma.)

4. There exists a constant $C > 0$ such that $\| J(\sum_{i=1}^{n} \alpha_{i} x_{i}) \|_{E^{**}} \leq C$ for all finite sequences $(\alpha_{i})_{i\in\mathbb{N}}$ in $\{0,1\}$ and $n\in\mathbb{N}$. (Applying that $J$ is linear.)

5. There exists a constant $C > 0$ such that $\| \sum_{i=1}^{n} \alpha_{i} x_{i} \|_{E} \leq C$ for all finite sequences $(\alpha_{i})_{i\in\mathbb{N}}$ in $\{0,1\}$ and $n\in\mathbb{N}$. (Applying that $J$ is an isometry.)