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**234578** So the question says to find all real eigenvalues and for each of these real eigenvalues find a real eigenfunction. V is the linear space of all functions

R->R.

a) T(f(x))=2f(x) now I can already see the only eigenvalue is 2 and any function with real coefficients works so f(x)=sinx is one such eigenfunction.

b) T(f(x))=f(-x)

Now the way I would usually approach this is I would find the matrix A=[T] with respect to the standard ordered basis of V and then use the characteristic polynomial f(t)=det(A-tI) BUT I can't figure out what the standard ordered basis would be. a basis for all polynomials would be (1, x, x^2,...) but in V, the space of ALL functions from R to R, functions like sinx are included so how can I find the matrix representation of T?

I know, for example, -1 and 1 are eigenvalues because T(sinx)=-sinx and T(cosx)=cosx but what about some function g(x)={1, when x>or=0 and g(x)={4, when x<0 ? wouldnt T(g(x))=4g(x) so can't any number be an eigenvalue of T?

Also T(h(x))=-(1/3)x if h(x)=abs(x)+2x so -1/3 is also an eigenvalue for T...so an any number be an eigenvalue? how do I show that?