# extension theorems on normed spaces

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 $\mathbb{R}$?

• Regarding the last question: note that a linear continuous function is automatically Lipschitz continuous. – Giuseppe Negro May 24 '14 at 15:49
• The extension is unique: if $D\ni d_n\to x$ and $f(x)$ exists and is continuous, than $f(d_n)\mapsto f(x)$. – Peter Franek May 24 '14 at 15:58
• @GiuseppeNegro Yes I forgot about that, so since it's Lipschitz continuous it is also uniformly continuous and therefore we use the usual Theorem regarding uniform continuity of a dense set? – user103184 May 24 '14 at 16:03
• @GiuseppeNegro Can I ask you A Sobolev space question. That's if you have any experience in that area? – user103184 May 24 '14 at 16:06
• If you have a question, please ask it on the main page rather than asking me personally. – Giuseppe Negro May 24 '14 at 16:09