Re: continous endomorphisms

eugene wrote:

E is a vectorial space of infinite dimension. Prove that there
is no continious endomorphisms f and g such that fg-gf=I.

How do you define "continuous" for general vectorial spaces?

On the other hand, if you don't assume continuity, then the statement
is false. Consider the space of all C^{oo}-functions from R into R
and define, when _h_ belongs to that space, f(h(x)) = h'(x) and
g(h(x)) = x*h(x). Then f o g - g o f = Identity.

Of course, the statement is true for finite-dimensional spaces. All
you have to do is to see that

0 = tr(f o g) - tr(g o f) = tr(f o g - g o f) = tr(Id),

which is equal to the dimension of the space.

Best regards,

Jose Carlos Santos
.

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