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What is $L^p(\{0,1\})$ ? i.e. a $L^p$ space on two points ? I'v never heard about it.

user352653
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1 Answers1

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That's the space of all functions $f\colon\{0,1\}\longrightarrow\mathbb R$, endowed with the norm$$\|f\|_p=\left(\bigl\lvert f(0)\bigr\rvert^p+\bigl\lvert f(1)\bigr\rvert^p\right)^{\frac1p}.$$Of course, it is basically $\mathbb{R}^2$ with the norm $\lVert\cdot\rVert_p$.

José Carlos Santos
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  • Oh I see. But why writing $L^p{0,1}$ instead of $(\mathbb R^2, |\cdot |_p)$ ? It's so confusing... But as the writing suggest, is it a "non convex" norm ? (but may be it's not a norm but maybe $d(x,y)=|x-y|_p$ is a "non convex" distance ?) – user352653 Sep 01 '18 at 22:54
  • @user352653 This is a particular case of the more general concept of $L^p(X)$, where $X$ is an arbitrary set. I don't know what a non-convex norm is. – José Carlos Santos Sep 01 '18 at 22:57
  • Does $L^p{0,1}$ is the same that $\ell^p{0,1}$ ? – user352653 Sep 01 '18 at 23:19
  • @user352653 Yes. I think that the notation $\ell^p$ is more common. – José Carlos Santos Sep 01 '18 at 23:22