  The trace-dev-div inequality in $H^s$ controls the trace in the norm of $H^s$
by that of the deviatoric part plus the $H^{s-1}$ norm of the divergence of a
quadratic tensor field different from the constant unit matrix. This is well
known for $s=0$ and established for orders $0\le s\le 1$ and arbitrary space
dimension in this note. For mixed and least-squares finite element error
analysis in linear elasticity, this inequality allows to establish robustness
with respect to the Lam\'e parameter $\lambda$.
