What is $L^p(\{0,1\})$ ? i.e. a $L^p$ space on two points?

Question

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

Answer 1

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 $\lVert\cdot\rVert_p$.