To paraphrase:
Prove that if as in norm then as in norm.
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Introduce a function and the constant function on .
Now the Cauch-Scwartz inequality tell us that:
or:
So there exists a such that:
But the sequence of positive terms on the right is a null sequence and hence the sequence of positive terms on the left is a null sequence, hence:
, as
Hence convergence in norm implies convergence in norm
CB
Consider the sequence of functions :
If I have set this up right this tends to the zero function in norm as goes to infinity. However it does not do so in norm.
You will need to confirm this in detail, as I have just roughted out the argument and there may be some remaining errors hanging about.
THis can be simplified to:
CB