# A Normed Vector Space Whose Dual Space Has No Uniformly Bounded Bases

With Hahn-Banach extension theorem, it can be showed that for a finite number of linear independent vectors $\{\|x_k\|\}$ there is a set of linear bounded functionals $\{f_l\}$ such that $f_l(x_k)=\delta_{kl}$.

I'm asked to think about the case when $\{\|x_k\|\}$ is countable.

Suppose $E$ is a normed vector space, and $\{x_k\}$ is a countable family of linear independent vectors of $E$, with $\|x_k\|=1, k=1,2,3,\cdots$. Is there an $E$, $E$ has no uniformly bounded linear functionals $\{f_k\}$, i.e. $\sup_{k\geq1}\{\|f_k\|\}<+\infty$, satisfying $f_k(x_l)=\delta_{kl}$ ?

• @Giuseppe Negro Thanks for your reminder. It was quite late when I posted this question. I typed in a hurry and did not pay attention to wording. And the answer is helpful, thanks a lot. I think the Chebyshev polymonial may also be used to show that the projection is unbounded. – SimonChan Nov 3 '16 at 14:22
• You are welcome. I removed my previous comment because it does not apply to the edited question. +1 (The linked answer is this one by Robert Israel) – Giuseppe Negro Nov 3 '16 at 16:31