I just think of one ... polynomials with T = d/dx.
Any more inventive ideas?
Suppose that T is a linear operator on the vector space V (over the ﬁeld
F ). Then for any natural number m, .
If V is ﬁnite-dimensional, then must happen for some natural number m.
Can you give me a counterexample for the case where V is inﬁnite-dimensional? (that is the image is decreasing in size for ever)
I came up with a one that is a little strange.Any more inventive ideas?
Consider, - an infinite tuple.
Okay, I am abusing notation but I think you have an idea of what I mean.
Now, is a vector space over in a natural way.
Define, by .
Notice that and .
Thus, is a linear transformation.
And in general, for .
Notice that .