prove that a sequence is bounded in the bidual

Question

let E be a normed vector space , let $x_{n}$ be a bounded sequence on E , prove that $x_{n}$ is bounded on the bidual E**.

Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. — Another User, Mar 10 '23 at 17:22
Duplicate: natural embedding of normed linear space is an isometry — Anne Bauval, Mar 10 '23 at 17:42

score -2 · Accepted Answer · answered Mar 10 '23 at 17:37

-2

Let $\{x_n\}\subset E$ be a bounded sequence, i.e. $\|x_n\|\le M$, for some $M>0$.

Then set $\varphi_n \in E^{**}$, defined as $$ \varphi_n(x^*)=x^*(x_n), \quad x^*\in E^*. $$ Then, for $\|x^*\|_*=1$ we have $$ |\varphi_n(x^*)|=|x^*(x_n)|\le \|x^*\|_*\|x_n\|=\|x_n\|\le M. $$ Then $\|\varphi_n\|\le M$, for all $n\in\mathbb N$.

answered Mar 10 '23 at 17:37

Yiorgos S. Smyrlis

83,933

1

We are not supposed to answer such no-effort questions. Moreover, it is a duplicate. – Anne Bauval Mar 10 '23 at 17:38
1

Thank you so much Mr. Smyrlis for your answer. – alex roverr Mar 10 '23 at 17:55

prove that a sequence is bounded in the bidual

1 Answers1