Let $X,Y$ be real normed vector spaces, and suppose that $T:X \to Y$ is a bounded linear operator with closed image.
Let $\tilde X,\tilde Y$ be the completions of $X,Y$, and let $\tilde T:\tilde X \to \tilde Y$ be the natural continuous extension of $T$ (which is unique).
Is $\text{Image } \tilde T$ closed in $\tilde Y$?
I proved that $\, \, \overline{\text{Image } \tilde T}^{\tilde Y}= \overline{\text{Image } T}^{\tilde Y} $, however, this doesn't seem to resolve the question. (We only know $\text{Image } T$ is closed in $Y$; it does not need to be closed in $\tilde Y$).