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.

  • $\begingroup$ 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

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