# Questions tagged [weakly-cauchy-sequences]

this tag is for questions about weak Cauchy sequences in the sense of weak topology on a normed linear space.

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### A possible characterization of WSC spaces

A Banach space $X$ is weakly sequentially complete (WSC) if every weakly Cauchy sequence in $X$ is weakly convergent. I will use the following classical result: Rosenthal's $\ell_1$ theorem: Every ...
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### What is the motivation behind the weak limit?

We defined the weak limit as: Let $X$ be a Banach space. $x_n \in X$ converges weakly to $x_0 \in X$, if $\: \:\forall _{\phi \in X^{*}}\:\phi \left(x_n\right)\rightarrow \phi \left(x_0\right)$ But ...
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### Weakly unconditionally Cauchy series in $X^*$

Let $X$ be a Banach space. A series $\sum x_n$ in $X$ is weakly unconditionally Cauchy (or weakly uncontionally convergent) if $\sum |x^*(x_n)| < +\infty$ for every $x^* \in X^*$. Exercise 3, page ...
For the following theorem. Let $S$ be a nonempty subset of $H$ and let $x:[0,+ \infty) \rightarrow H$. Assume that $\quad$ (i) for every $z\in S$, $\lim_{t\rightarrow \infty} \left\|x(t)-z\right\|$ ...