What is the difference between a *surjective* unitary operator and a *bijective* unitary transformations

Question

The wikipedia page on unitary operators states they are surjective, whereas the page on unitary transformations states that they are bijective.

What is the difference between them and when is it correct to use one instead of the other? In my specific case I am interested in unitary operations in the context of quantum mechanics.

Answer 1

You are not reading everything it says. The first sentence says that $U$ is unitary if it is surjective and it preserves the inner product. This last condition implies (it is actually equivalent to) that $$\|x\|^2=\langle x,x\rangle=\langle Ux,Ux\rangle=\|Ux\|^2.$$ So $U$ is an isometry and thus injective.

Note that "bijective" implies "surjective".