  We show that certain symmetric seminorms on $\mathbb{R}^n$ satisfy the
Leibniz inequality. As an application, we obtain that $L^p$ norms of centered
bounded real functions, defined on probability spaces, have the same property.
Even though this is well-known for the standard deviation it seems that the
complete result has never been established. In addition, we shall connect the
results with the differential calculus introduced by Cipriani and Sauvageot and
Rieffel's non-commutative Riemann metric.

Functional Analysis