I know that there are a number of extension theorems, Tietze's extension theorem, Hahn-Banach extension and so on..
I want to know if there is an extension theorem which guarantees that if say $X$ is a normed space with a dense subset $D \subset X$, then taking some $f \in D^{*}$, there is an extension $g \in X^{*}$? Is it a unique extension? If there is an extension why is it enough for $f$ to be continuous and not uniformly continuous as is in the case for the real valued function $f:D \rightarrow \mathbb{R}$ on some dense subset of $\mathbb{R}$?