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Let $X$ be a normed vector space and $X_1,X_2$ be two dense subspace of $X,$ in Is is true that the intersection of two dense subspaces of a linear normed space is also dense? it was showed by an example that $X_1\cap X_2$ is not necessarily dense in $X,$ but in that example $\mathrm{codim}X_1=\mathrm{codim}X_2=+\infty.$ So I wonder if $\mathrm{codim}X_1$ or $\mathrm{codim}X_2$ is finite, will the intersection $X_1\cap X_2$ be dense in $X$ ?

dsh
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Tiffany
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    Take $f_1$ continuous and $f_2$ discontinuous linear functionals. Then $f_1+f_2$ and $f_2$ are dense of codimension 1 whose intersection is $\ker f_1$ closed of codimension 1. – dsh Nov 04 '23 at 10:44
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    @dsh You mean the kernels of $f_1+f_2$ and $f_2$. Nice example. – Jochen Nov 04 '23 at 10:58
  • @Jochen Yes, I meant kernels. I guess I have seen this example before. – dsh Nov 04 '23 at 11:01
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    @dsh A nice example! Though the intersection should be $\ker f_1\cap\ker f_2$ actually... – Tiffany Nov 04 '23 at 11:03
  • @Tiffany Thank you for correction! I cannot modify comment, unfortunately. – dsh Nov 04 '23 at 11:08

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