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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