Reference request: $T$ bounded operator from Hilbert space to itself, then $\text{Ran}\,T$ closed equivalent to $\text{Ran}\,T^*$ closed.

Question

Where can I find a proof of the following fact?

Let $T$ be a bounded operator from $H$ to $H$, with $(H, (\cdot, \cdot))$ a Hilbert space. Then $\text{Ran}\,T$ closed is equivalent to $\text{Ran}\,T^*$ closed.

Answer 1

You can show that an injective linear map between Banach spaces is closed if and only if it is bounded below. I think you can use the fact that $T$ and $T^*$ have the same norm that they are both bounded below if one is.