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.

  • $\begingroup$ 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. $\endgroup$ – Michael Hardy May 4 '13 at 0:30
  • 2
    $\begingroup$ @felipeuni : sorry to nitpick, but the word is "supremum". $\endgroup$ – Stefan Smith May 4 '13 at 0:53

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.