3

Let $X$ be the normed vector space $C^1[0,1]$, of continuously differentiable functions on $[0,1]$ with the sup norm $\displaystyle \|f\|=\max_{t\in[0,1]}|f(t)|$. Find a sequence of bounded linear functionals $T_n:X\to\mathbb{R}$ such that for every $f\in X$, $\displaystyle \sup_{n\in N}|T_nf|\lt\infty$ but $\displaystyle \sup_{n\in N}\|T_n\|=\infty$. (That is to say, the principle of uniform boundedness fails if $X$ is not complete).

I have tried looking at a few $T_n:X\to\mathbb{R}$ but I seem rather stuck at finding an example. Some of those which I have tried are: $T_n=n\|f\|$ (this one didn't work as $\sup|T_nf|=\infty$), $T_n={\|f\|\over n}$ (this one didn't work because $\sup\|T_n\|=1$)...

Any suggestions on how to come up with this example?

Rudy the Reindeer
  • 46,359
user3784030
  • 1,013

1 Answers1

2

Consider $T_n f=f'(1/n)$. Since $f\in C^1$, $|T_n f|\leq \sup |f'(x)|<\infty$. On the other hand, $f_t(x)=e^{tx}$ is a $C^1$ function and $T_nf_t=te^{t/n}\rightarrow t$ as $n\rightarrow\infty$.

Edit: I just noticed that you want the functionals to be bounded. In this case, consider instead $$ T_n f=n(f(1/n)-f(0)). $$ We see that $T_nf\rightarrow f'(0)$, so $|T_nf|$ is bounded. On the other hand, $\|T_n\|= n$.

Jonas Dahlbæk
  • 4,776