# Effect of dualization of density

Let $$D\subset X$$ be a dense subset of a complete separable locally convex space $$X$$ over $$\mathbb{R}$$. Though the question seems simple enough, I can't seem to find the answer in the literature:

• If $$D$$ is a dense subspace of $$X$$, then do the continuous duals $$D'$$ and $$X'$$ coincide?

• More generally, if $$D$$ is not a linear subspace, can we give any meaning to its "dual" $$D'$$ and if so, how does this object relate to $$X$$?

• For $D$ a subspace this seems immediate to me, for $X$ an arbitrary topological vector space. Use Cauchy nets (and completeness of $\mathbf{R}$).
– YCor
Jun 4, 2021 at 8:36
• The second question ($D$ not a subspace) is quite speculative. For instance, in an infinite-dimensional Hilbert space, $D$ might be a subset that is linearly independent over $\mathbf{R}$.
– YCor
Jun 4, 2021 at 8:40
• As YCor remarks, the first statement is well-known and quite simple (one does not need completeness or separability of $X$). Instead of the extension by uniform continuity one can also use Hahn-Banach for the extension which is unique by density. Jun 4, 2021 at 8:56
• @YCor Interesting point, in that case I guess I can reduce to the first and just consider the span in my setting. Let me ask, is there a good reference where I can find the first result? Jun 4, 2021 at 11:25
• But Hahn-Banach is false for general (Hausdorff) topological vector spaces. And in the locally convex setting anyway it sounds curious to use a non-constructive theorem to get a unique extension. In any case, the use of Zorn's lemma in the proof has a step which is that the maximal subspace on which the extension is defined, has to be closed. This eventually makes use of this immediate Cauchy argument.
– YCor
Jun 4, 2021 at 12:35