[mistaken]
Let be defined by
Let and
Show that with respect to , but is not convergent with respect to
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Now, I did the following, but it must be wrong since it doesn't depend on the choice of norm, so I get that in both cases.
Choose for some small n.
st but rather
st but rather
st
So, clearly, we have that st
So what have I missed that stops the above from working, and what is a better way of doing it?
Why not just compute and ? You can notice that hence
for . For , if you get 0, you're done. For , you should not get 0; if you get an infinite limit, there is no limit in . If there is a finite limit... it may need some more work to find it or prove there is none. (I think here you get an infinite limit, just because of the first term , so you're done)
In fact, since the space is , if in , then in hence the limit should be 0 and it suffices to check if to know if there is convergence in .
What you proved here is that converges to almost-everywhere (in fact, everywhere). As you probably know, you may have for almost all (or all) , and . This is however true under a few conditions: cf. the bounded convergence theorem. In this case it is not straightforward to apply it to prove convergence in , and anyway it wouldn't suffice for the case.Now, I did the following, but it must be wrong since it doesn't depend on the choice of norm, so I get that in both cases.
Choose for some small n.
st but rather
st but rather
st
So, clearly, we have that st