Consider a subspace $M$ of a complex normed vector space $(X,\|\cdot\|)$, $p:X\to\mathbb{R}$ sublinear and $f:M\to\mathbb{C}$ linear such that $|f(x)|\le p(x)$ $\forall x\in M$. We want to prove that there exists $\Lambda:X\to\mathbb{C}$ such that $\Lambda|_{M}=f$ and $|\Lambda(x)|\le p(x)$ $\forall x\in X$.
Write $$f(x)=\Re f(x)+i\Im f(x)\qquad\text{for }x\in M$$ Since $f(ix)=if(x)$, we see that $$\Re f(ix)+i\Im f(ix)=i\Im f(x)+i\Re f(x)$$ Thus $\Im f(x)=-\Re f(x)$. Therefore $$f(x)=\Re f(x)-i\Re f(ix)$$
$\Re f:X\to\mathbb{R}$ is a continuous real linear functional on $X$, so I want to apply the (real) Hahn-Banach theorem to it. However, I am having struggle bounding $\Re f(x)$ by $p(x)$.
