## Jordan-von Neumann Theorem and Lp spaces

$L^{p}$ spaces crept into my mind earlier today, so I thought I would write a brief post about them. A while ago, I posted an exposition of the well-known Jordan-von Neumann theorem, which essentially states that a normed space $(X,\left\|\cdot\right\|)$ has an inner-product which induces $\left\|\cdot\right\|$ if and only if the parallelogram law

$\displaystyle\left\|x+y\right\|^{2}+\left\|x-y\right\|^{2}=2\left(\left\|x\right\|^{2}+\left\|y\right\|^{2}\right),\indent \forall x,y\in X$

hods. We can use the Jordan-von Neumann theorem to show that $L^{p}(\mathbb{R}^{d})$ is an inner-product space if and only if $p = 2$. The $\Leftarrow$ direction is clear, so we concern ourselves with the $\Rightarrow$ direction. Define measurable sets $E_{1} := [0,1]^{d}$ and $E_{2} := [1,2] \times \prod_{j=1}^{d-1}[0,1]$, for $\epsilon > 0$ small, and let $\mathbf{1}_{E_{1}}$ and $\mathbf{1}_{E_{2}}$, respectively, be the corresponding characteristic functions. Since $E_{1}$ and $E_{2}$ are almost disjoint, we have that

$\displaystyle \begin{array}{lcl}\left\|\mathbf{1}_{E_{1}} + \mathbf{1}_{E_{2}}\right\|_{L^{p}}^{2} + \left\|\mathbf{1}_{E_{1}} - \mathbf{1}_{E_{2}}\right\|_{L^{p}}^{2} &=& \left(\int_{\mathbb{R}^{d}}\left|\mathbf{1}_{E_{1}} + \mathbf{1}_{E_{2}}\right|^{p}\right)^{\frac{2}{p}} + \left(\int_{\mathbb{R}^{d}}\left|\mathbf{1}_{E_{1}} - \mathbf{1}_{E_{2}}\right|^{p}\right)^{\frac{2}{p}}\\&=& 2\cdot 2^{\frac{2}{p}}\end{array}$

which equals $4 = 2(\left\|\mathbf{1}_{E_{1}}\right\|_{L^{p}}^{2} + \left\|\mathbf{1}_{E_{2}}\right\|_{L^{p}}^{2})$ if and only if $p =2$.

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