Extension of a linear map from a one dimensional subspace to its closure

Question

Let $F$ be a linear subspace of a normed space $E$. Let $G$ be a Banach Space. Prove that

1. For any continuous linear map $T:F\rightarrow G$, there exists exactly one continuous map $\overline{T}:\overline{F}\rightarrow G$ such that $\overline{T}\mid_{F}=T$.

2. The map $\overline{T}$ is linear.

3. $||\overline{T}||=||T||$