beginning of the proof of fundamental solution of Poisson's equation - Evan's PDE

Question

in Evan's PDE book (see pic below)

$D^2f$ being the hessian matrix and $\Delta f = Trace(D^2f)$

I can't see why the highlighted equality in green hold ?

for example if you take $f(x,y) = x^2+y^2$

$D^2f = 2I_d \implies \| D^2f \|_{\infty} = 2$

but $\Delta f = 4 > \| D^2f \|_{\infty} $

so is $\| D^2f \|_{L^{\infty}(\mathbb{R^n})}$ something totally different from $ \| D^2f \|_{\infty}$ ?