1
$\begingroup$

I need a proof for special case of Riesz lemma (when $\varepsilon$ is 0):

If Y is a closed proper subspace of $L^p(\mu)$ for some $1<p<\infty$, then there exist $f\in L^p(\mu)$ such that $||f||=1$ and $||f-g||\geq 1$ for every $g\in Y$.

I know that Clarkson's inequality (uniform convexity) can be used.

$\endgroup$
  • $\begingroup$ This is an easy application of Hahn-Banach (hint: Every element of the dual of $L^p$ for $1<p<\infty$ attains its norm...) $\endgroup$ – David C. Ullrich Jan 12 at 13:06
  • $\begingroup$ Can you give me further explanation how to apply that? $\endgroup$ – Hana Jan 16 at 11:08

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.