Research proof index

Hölder's inequality, binary case.

Translation preserves the compact normalized $L^p$ norm: for every group element $a$, $\|\tau_a f\|_{p}=\|f\|_{p}$.

The discrete normalized $L^p$ norm is unchanged by conjugating a function and reflecting its argument: $\|x\mapsto\overline{f(-x)}\|_p=\|f\|_p$.

For nonzero natural exponents $p$ and $q$, taking a pointwise $q$-th power rescales the discrete normalized norm by $\|f^q\|_p=\|f\|_{pq}^{q}$.

Hölder's inequality, finitary case.

Translation preserves the discrete normalized $L^p$ norm: for every group element $a$, $\|\tau_a f\|_{p}=\|f\|_{p}$.