  We survey some classical norm inequalities of Hardy, Kallman, Kato,
Kolmogorov, Landau, Littlewood, and Rota of the type \[ \|A f\|_{\mathcal{X}}^2
\leq C \|f\|_{\mathcal{X}} \big\|A^2 f\big\|_{\mathcal{X}}, \quad f \in
dom\big(A^2\big), \] and recall that under exceedingly stronger hypotheses on
the operator $A$ and/or the Banach space $\mathcal{X}$, the optimal constant
$C$ in these inequalities diminishes from $4$ (e.g., when $A$ is the generator
of a $C_0$ contraction semigroup on a Banach space $\mathcal{X}$) all the way
down to $1$ (e.g., when $A$ is a symmetric operator on a Hilbert space
$\mathcal{H}$).
  We also survey some results in connection with an extension of the
Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt.
