what is the dual space of this vector space $W=\{(x_1,\dots,x_n)|x_1+\dots+x_n = 0\}$

Question

I have this question in my linear algebra homework, but I can't wrap my head around it. We're supposed to prove that $$ W^* = \{f|f(x_1,\dots,x_n) = \sum_{i=1}^n a_ix_i,\sum_{i=1}^na_i=0\} $$ But I can't reason to myself why for example the function $f(x_1,\dots,x_n) = \sum_{i=1}^n x_i$ is not in $W^*$. Is it not a linear function that assigns a scaler to every vector in $W$?