Hi for all,,
Can someone prove that:
If A is an additive operator and continuous from a linear space X into a linear space Y;then A is a linear operator ??!
thanks for all respondings
The spaces here should have a topology, or continuity is pointless.
So, consider them to be real topological vector spaces (which is consistent with the fact that we need to have a linear map here).
additive (i.e. ) and continuous.
To show is linear, we show that for , we have (%).
We easily see that , for (*). Also, we notice that by additivity , so . So (*) also holds for negative n.
For , we have . Now (*) grants us (%) for the rationals.
For , consider a sequence of rationals We have
. Now and , thus
Thus the norm induces a metric space.
You did a good job is showing the standard properties.
Having graded this sort of proof for years, I did wonder if you were sure of that last sentence.
Suppose that that means: .
Do you see the contradiction?
Oops. In the books I used to read, linear space just went for vector space.In my experience the term linear space means a normed linear space.
Which one? That looks good in a TVS?if you were sure of that last sentence.
In a metric space, your thingie is much more neat.
In a TVS, the fun begins