Holder's inequality with expectation norm One defines the ``p-expectation norm" of a function $f$ as $\vert f \vert_p = (\mathbb E (  f^p) )^{ \frac{1}{p}}$. 


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*What is the intuition for this norm? 


Now why are the following true? 


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*$\vert x \vert_q = max_{\vert y \vert_{q/(q-1) } \leq 1 } \vert <x,y> \vert$

*$\vert <x,y> \vert \leq \vert x \vert _ {q/(q-1) } \vert y \vert _q$



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*You can imagine $x$ as being defined over finite sets and hence there are no convergence issues here. 

*I think in the above it means that $\vert  <x,y> \vert = \vert E ( xy ) \vert$
 A: Your definition is missing an absolute value: it should be $\|f\|_p = (E(|f|^p))^{1/p}$. Of course, you can drop the absolute values if $f$ is real and nonnegative.
The intuition for the $p$ norm is that it is a generalization of the familiar norms $\|f\|_1 = E(|f|)$ and $\|f\|_2 = (E(|f|^2))^{1/2}$. Larger values of $p$ assign more weight to points where $|f|$ is large, and less weight to points where $|f|$ is small. The case $p=\infty$ is a limiting case. 
One reason to use a $p$-norm other than $p=1$ or $p=2$ is to deal with functions for which $\|f\|_1$ or $\|f\|_2$ is infinite, but $\|f\|_p < \infty$ for some other values of $p$. For example, if $f(x) = 1/x^{1/2}$ (assuming, say, Lebesgue measure on $[1,\infty)$, then $\|f\|_1 = \|f\|_2 = \infty$, but $\|f\|_p < \infty$ for all $p > 2$.
For the special case of $p=2$, the Cauchy-Schwarz inequality holds:
$$|\langle f,g \rangle| \leq \|f\|_2 \|g\|_2$$
The Hölder inequality generalizes the Cauchy-Schwarz inequality to arbitrary $1 \leq p \leq \infty$:
$$|\langle f,g \rangle| \leq \|f\|_p \|g\|_q$$
where $q$ is the number satisfying $1/p + 1/q = 1$, so $p = \frac{q}{q-1}$ and $q = \frac{p}{p-1}$. This immediately gives us your desired inequality,
$$|\langle x,y\rangle| \leq \|x\|_{q/(q-1)}\|y\|_q$$
To get the equation involving the max, start by interchanging $x$ and $y$ so we have
$$|\langle x,y\rangle| \leq \|x\|_q \|y\|_{q/(q-1)}$$
If $\|y\|_{q/(q-1)} \leq 1$ then this simplifies to
$$|\langle x,y\rangle| \leq \|x\|_q$$
The Hölder inequality (like the Cauchy-Schwarz inequality) becomes an equality in the special case where $x,y$ are scalar multiples of each other. Therefore, the left hand side achieves the maximum value $\|x\|_q$ when $y$ is any scalar multiple of $x$ satisfying $\|y\|_{q/(q-1)} \leq 1$, and this gives the desired result:
$$\max_{\|y\|_{q/(q-1)} \leq 1} |\langle x,y\rangle| = \|x\|_q$$
