Reading the proof of an article of E. Lieb I couldn't understand the following: for $0<p<\infty$ and $g\in L^1(\mathbb{R}^n) \cap L^p(\mathbb{R}^n)$, $\|g\|_1=\sup\lbrace\int_{\mathbb{R}^n}f(x)g(x)dx\vert f\in L^q(\mathbb{R}^n) \cap L^\infty(\mathbb{R}^n), \|f\|_1<1\rbrace$, where $q$ is the conjugate exponent of $p$. By the dual characterization of the norm, we know that $\|g\|_1=\sup\lbrace\int_{\mathbb{R}^n}f(x)g(x)dx\vert f\in L^\infty(\mathbb{R}^n),\|f\|_1\leq 1\rbrace$, but $L^q(\mathbb{R}^n) \cap L^\infty(\mathbb{R}^n)$ is not dense in $L^\infty(\mathbb{R}^n)$. I would be grateful for any help.

  • 2
    $\begingroup$ Just truncate any given $ L^{\infty}$ function from the supreme characterization that you already know $\endgroup$ – Bananach Jul 28 '17 at 17:07
  • $\begingroup$ Could you please clarify what part of this statement you don't understand? $\endgroup$ – Michael Lee Jul 29 '17 at 7:18
  • $\begingroup$ I don't understand the first equality. The second part ("by the dual...") was just my attempt $\endgroup$ – tigro Jul 29 '17 at 14:02

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