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I was wondering about the following:

Let $X$ be a normed space and $A$ be a dense subset in $X$. What properties must a function $f$ necessarily have, so that we have that $f(A)$ is dense in $Y$, if we have $f:X \rightarrow Y$. I think it is pretty clear that the function should be bijective? Do we need more(continuous?)

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  • $\begingroup$ Do you want the image of every dense subset to be dense, or just the one? $\endgroup$ – Henno Brandsma Dec 28 '13 at 14:35
  • $\begingroup$ of every dense subset $\endgroup$ – user66906 Dec 28 '13 at 15:04
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Not sure about necessary conditions. But if $f$ is continuous and $f[X]$ is dense in $Y$, you already know that $f[A]$ is dense in $Y$, by standard arguments. So bijective is not necessary. Of course $f[X]$ must be dense (as it contains $f[A]$).

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  • $\begingroup$ what do you mean by $f[Y]$? $\endgroup$ – user66906 Dec 28 '13 at 15:13
  • $\begingroup$ Corrected to $f[X]$ $\endgroup$ – Henno Brandsma Dec 28 '13 at 15:16

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