What is the dimension of eigenspaces in a function-based linear transformation?

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Linear transformation f:C^∞(R) -> C^∞(R)

f(x(t)) = x'(t) a) I have to set up the eigenvalue-problem and solve it :

My solution : ke^λtb) Now I have to find the dimension of the single eigen spaces when λ is

-5 and 0. My solution :

Eigenspaces :

E-5 = ke^-5t

E0=k (because ke^0t = k)

But I don't know how to find the dimension of the single eigen spaces ?

I'm used to working with vectors but now it's functions and I'm not sure about the dimension.
 
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Good! I presume that you realized that since every solution, Ce^{\lambda x} is a constant, C, times the single function e^{\lambda x}, the space is one dimensional.
 

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