4
$\begingroup$

Here $C(\mathbb R)$ is the set of all continuous functions in $\mathbb R$, so the problem asks me to prove that there exists $f \in L^\infty(\mathbb R)$ such that there is no sequence of functions in $\mathbb R$ converging under sup norm to $f$. Also, the continuous functions not necessarily have compact support.

Does anyone have ideas? Thank you for your time.

$\endgroup$
  • 1
    $\begingroup$ Hint: Convergence in the sup norm is also known as uniform convergence. Do you know any results concerning uniform convergence of continuous functions? $\endgroup$ – mlk Jul 11 '17 at 13:29
1
$\begingroup$

Take the function $f(x)=1_{[0,1]}+1_{[2,3]}$.We have that $f \in L^{\infty}$ and $f$ is not continuous.

If there was a sequence of continuous functions converging uniformly(under the sup norm) to $f$ then $f$ would be continuous,which is a contradiction.

$\endgroup$

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.