You can prove that $T$ is diagonalizable if and only if the minimal polynomial is separable and splits in your field. For you that means that you need to find the minimal polynomial for $T$ and show it has all real distinct roots.
Let f1(t)=e^t, f2(t)=te^t, f3(t)=t^2e^t, and let V=Span(f1,f2,f3) in the infinite continuous functions. Let T:V-->V be give by T(f)=f''-2f'+f. Decide whether T is diagonalizable.
We learned a theorem that this will be diagonalizable if and only if the geometric multiplicity of each eigenvalue equals its algebraic multiplicity.
What I am having trouble with is translating this into a way to find geometric and algebraic multiplicity. I'm not entirely sure what to do when I'm not given a matrix, since that's how we did it in class.
I have to do it using what we have learned so far. We haven't learned that yet, and I don't even know how to find the characteristic polynomial of this function, so I'm not sure how to find a minimal polynomial. Could you help me with finding a polynomial for T? We've only used matrices to do it, and I"m not sure how to put this into a matrix to find the eigenvalues.