Is surjective strictly contracting linear map between Hilbert spaces a bijection?

Question

Suppose $U$ and $V$ are Hilbert spaces and $T:U\rightarrow V$ is a surjective continuous linear map with $0<\|{T}\|<1$. How to prove $T$ is injective?

Answer 1

If this were true then any bounded surjective operator would be injective: Say $T$ is bounded and surjective. Then $T\ne0$, so $||T||>0$. If $a>0$ is small enough then $||aT||<1$, so $aT$ is injective, hence $T$ is injective.