In a recent work, Nathanson proved that, for $K=\mathbb {R}$ or $\mathbb {C}$, the set of $B_h$-vectors in $K^n$ is open and dense. Leonetti later extended this to $B_h[g]$-vectors. In this note, we isolate a finite linear functional principle behind these results. Namely, if $\mathcal F$ is a finite family of pairwise distinct linear functionals on $K^n$, then the set of points at which no value is assumed by more than g members of $\mathcal F$ is open and dense in $K^n$. As consequences, we recover the theorems of Nathanson and Leonetti for $B_h[g]$-vectors and obtain the following variant: for every linear form $L(x_1,\ldots ,x_h)=c_1x_1+\cdots +c_hx_h$, with nonzero integer coefficients, the corresponding set of $L[g]$-vectors, evaluated on distinct variables, is open and dense in $K^n$.