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I was given the following conditions in an exercise:

  • Define an $L^p (R,dx)$ space, for $p \geq 1$ as a space of the class of equivalence of functions $f$ that satisfies the condition: $$\int {\vert f(x) \vert}^p dx < \infty $$
  • For such functions, the p-norm is defined as follows: $$ {\vert \vert f \vert \vert}_p = {\left[ \int {\vert f(x) \vert}^{p} dx \right]}^{1/p}$$

Find the values of p for which ${\vert \vert \cdot \vert \vert}_p$ derives from an inner product, then give the explicit forms of those inner products.

Attempt at solution: So my first thought was that if I manage to prove that the parallelogram law holds for a given norm, then that norm was induced by an inner product. After that, I could possibly find the explicit form of the inner product using the polarization identity, but the question still is how do I find the values of p? I clearly have no idea how to approach that.

Edit: Thinking about how to approach this, is it possible to use characteristic functions from some intervals, say, from $[0,1]$ and $[1,2]$ for instance? That way I could just pin-point the norms that are actually induced from inner products. Could it be easier that way?

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A standard way of working with these things is to just test against characteristic functions. Take $A$ and $B$ disjoint sets so that the $L^p$ norms of the characteristic functions are $|A|^{1/p}$ and $|B|^{1/p}$ respectively.

Now write down the parallelogram law applied to $\chi_A$ and $\chi_B$. You get (spoilers!) that

$$2 |A|^{2/p} + 2 |B|^{2/p} = 2 |A \cup B|^{2/p} = 2 (|A| + |B|)^{2/p}.$$

It should be pretty quick from here to deduce the (one and only) correct value of $p$.

The value is $p = 2$, unsurprisingly, since this is induced by the inner product $\langle f, g \rangle = \int f \overline{g} \, d\mu$.

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