Let $X$ be a real normed linear space, and $l$, $l_1$ linear functionals on a subspace $Y$ of $X$, such that $$|l(y)|+|l_1(y)|\le \|y\|, \quad y\in Y.$$
I would like to show that there exists $L, L_1 \in X^*$ that extend $l$ and $l_1$, respectively, and also $$|L(x)|+|L_1(x)|\le \|x\|, \quad x\in X.$$
I know that the extension can be made by the Han-Banach theorem, since $l, l_1$ are bounded functionals. However, I am not sure how I can guarantee that the dominance condition on the whole vector space still holds. Any thought?