\documentclass{minimal}
\usepackage{amsmath}
\usepackage[a6paper]{geometry}
\begin{document}

\newcommand\norm[1]{\left\lVert #1 \right\rVert}

$$ \norm x + \norm p + \norm{p^p} + \norm{p^{p^p}} + \norm{p_p^p} $$

$$ \norm{p_p^p} + \norm{Q_Q^Q} + \norm{x_x^x} $$

$$ x^{dp} x_d^p x_p^d x^{pd} $$

$$ \left\lVert x_d^p \right\rVert < \left\lVert x_i^p \right\rVert $$

$$ \left\lVert x_i^a \right\rVert < \left\lVert x_i^p \right\rVert $$

\end{document}