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)
Thanks.
Why does this work?
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.
Define, .
Define, .
Define, .
And in general, for .
Notice that .
However, .
Thus, .
In operator theory, this is called the unilateral shift. It is the device used by the manager of the Hilbert Hotel to ensure that an empty room can always be guaranteed, no matter how many guests arrive.