So we were emailed the following: Given P is the set of polynomials with a degree at most n is a vector space with dimension n + 1, give an example of a element not in the image of the second derivative operator, f(x)'':P20 ->P20.
Am I giving something that its second derivative doesn't exist, like a 2x + 1, or am I giving a polynomial that its second anti derivative would not be in P20, like 3x^20, its second derivative will not be in the original domain. He said he couldn't give any more hints....I just need to know which one he wants.
The second derivative of $f(x)=2x+1$ certainly exists. It's $0$.
Your second approach is exactly right. You want a polynomial of degree at most $20$ that is the second derivative of some polynomial not of degree at most $20$.