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.
9
questions
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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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6
votes
1
answer
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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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0
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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 ...
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The difference between 'weak limit point' and 'converge weakly'
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 $z\in S$, $\lim_{t\rightarrow \infty} \left\|x(t)-z\right\|$ ...
- 26
0
votes
2
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bounded $c_0$ sequence has weakly cauchy subsequence
I'm solving Conway's Functional Analysis. (Weakly Compact Operator)
The problem is
Show that every bounded sequence in $c_0$ has a weakly Cauchy subsequence, but not every weakly Cauchy seqence in $...
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2
votes
1
answer
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About weakly Cauchy sequences in complete metric spaces
Let $(X,d)$ be a complete metric space. Call a sequence $(x_n)\subseteq X$ a weakly Cauchy sequence in $X$ if there is some $y\in X$ such that $(d(y,x_n))_n$ is a Cauchy sequence in $\mathbb{R}$. It ...
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weakly cauchy sequence is norm compact
A sequence $(x_n)$ is weakly cauchy if for every $x^*\in X^*$, $(x^*(x_n))$ converges. Let $c$ denote the space of convergent functions.
Theorem: A weakly cauchy sequence is norm-bounded.
Proof: Let ...
- 2,991
0
votes
2
answers
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Space of weakly Cauchy sequences
Let $X$ be a Banach space and let us consider the linear subspace of $\ell_\infty(X)$ comprising all weakly Cauchy sequences.
Is this subspace closed?
It is not as immediate as in the case of ...
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4
votes
1
answer
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Weakly Cauchy sequences need not be weakly convergent
A sequence $(x_n)$ in a Banach space $X$ is called weakly Cauchy if for every $\ell \in X'$ the sequence $(\ell(x_n))$ is Cauchy in the scalar field.
I want to show that weakly Cauchy sequences ...
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