Show that every sequence weakly cauchy is bounded
I don't know here how to use the hypotheses that $x_{n}$ is a weakly cauchy sequence. The definition i have is:
A sequence $\{x_{n}\}$ in a normed space E is called a weakly cauchy sequences if for every $f \in E^{*}$, the sequence $\{f(x_{n})\}$ is a cauchy sequence in $R$.
Best Answer
Consider the family of maps $$T_{n} :X^{*} \to \mathbb{R}, T_{n} (f) =f(x_{n}) $$, Since $f(x_{n}) $ is Cauchy for every fixed $f$, it is bounded for every $f$, So the family of linear maps defined above is pointwise bounded, we can apply Banach-Steinhaus theorem, to ensure the existence of positive constant $M>0$ such that $\sup ||T_{n} ||<M$, but we know that ||T_{n} ||=||x_{n} || (this can be proved using Hahn-Banach theorem), So the sequence $x_{n} $ is bounded.