It seems to me I saw a counterexample somewhere, but I can't find it, can anybody help me?
Let $\varphi:X\to Y$ be a linear continuous map of locally convex spaces, and $\widetilde{\varphi}:\widetilde{X}\to\widetilde{Y}$ the corresponding linear continuous map of their completions.
If $\varphi:X\to Y$ is injective, is $\widetilde{\varphi}:\widetilde{X}\to\widetilde{Y}$ injective as well?
I think, the answer must be "no", so another question is
Under which conditions the injectivity of $\varphi:X\to Y$ implies the injectivity of $\widetilde{\varphi}:\widetilde{X}\to\widetilde{Y}$?