I am preparing for my exam right now and I am solving questions. I proved something but I am not sure. I have uploaded the question with the definitions just in case we have different notations.

So the questions is:

V and W are vector spaces. f is a mapping from V to W. f is injective if and only if f dual is surjective.

(I have solved something really similar to this so this is why I came up with the solution but still I am not sure.)

The file is more detailed. Appreciate the feedbacks. Thank you.