SCHWEITZER 2017 Problem 9

Let $N$ be a normed linear space with a dense linear subspace $M$ . Prove that if $L_1,\ldots,L_m$ are continuous linear functionals on $N$ , then for all $x\in N$ there exists a sequence $(y_n)$ in $M$ converging to $x$ satisfying $L_j(y_n)=L_j(x)$ for all $j=1,\ldots,m$ and $n\in \mathbb{N}$ .