7 5 points Bonus Find an example and justify your answers of

7. (5 points) Bonus: Find an example (and justify your answers) of an infinite dimensional vector space V and two linear maps f, g: V V such that (a) f is injective, but not surjective and (b) g is surjective, but not injective. As we will see in class, this is only possible for infinite dimensional vector spaces.

Solution

a) f is injective, because if g1g1 and g2g2 differ at some point -- say g1(x0)g2(x0)g1(x0)g2(x0), then f(g1)f(g1) and f(g2)f(g2) will have different derivatives at that point, and so cannot be the same function. On the other hand ff is not surjective because, for example, the function xx+1xx+1 is not f(g)f(g) for any gg (namely f(g)(0)=00g(t)dt=00+1f(g)(0)=00g(t)dt=00+1)