Give an example of a continuous linear operator $\displaystyle\|T\|=\sup_{\|x\|\le1} \|T(x)\|$ such that the supremum not reached

Question

Let $T:X\longrightarrow Y$ be a continuous linear operator , $X \;,\;Y$ normed spaces with

$$\|T\|=\sup_{\|x\|\le1} \|T(x)\|$$

Give an example of a continuous linear operator such that the supremum not reached

$$\|T(x)\|<\|T\|\;\; ,\;\; \|x\|\le 1$$

If the space $X$ is finite dimensional the unit ball is compact then the supremum is reached

Any hints would be appreciated.

Please. Don't write \underset{xxx} {\text{sup}}. Write \sup_{xxx}. In a "displayed" setting, that yields the same result, except that it also automatically provides proper spacing before and after "$\sup$" in things like $a\sup b$. And it's standard. — Michael Hardy, May 04 '13 at 00:30

score 8 · Accepted Answer · answered May 04 '13 at 00:33

8

$$ T:\ell_2\to\ell_2:(x_n)\mapsto((1-2^{-n})x_n) $$

answered May 04 '13 at 00:33

Norbert

1 Answers1