Suppose [itex]T[/itex] is an injective linear operator densely defined on a Hilbert space [itex]\mathcal H[/itex]. Does it follow that [itex]\mathcal R(T)[/itex] is dense in [itex]\mathcal H[/itex]? It seems right, but I can't make the proof work... There is a theorem that speaks to this issue in Kreyszig, and also in the notes provided by my professor; however, my professor's notes seem to indicate that the answer to the above question is "YES", whereas Kreyszig seems to indicate that it's "NO". Oh...and if micromass reads this, then: Thank you, so much, for helping me out over the past several days.