Dense Subspaces of Sobolev Spaces
Let . Then is called the -th weak derivative of iff
for all with compact support in .
Now for consider the Sobolev space of functions which have weak derivatives up to order ,
endowed with the norm
Consider also for the subspace in , which consists of all real-valued m-times continuously differentiable functions on with weak derivatives
of all orders up to , i.e. .
There is a theorem which says that the closure with respect to the norm above is exactly for all , so that is the exception.
Before it is shown by example that for one inclusion is not fulfilled it is said that
with respect to the above norm with .
This is clear because is complete.
The author then unexpectedly concludes that with respect to
While switching to m=1 he apparently simultaneously shrinks the class of functions on the left, but the norm with respect to which the closure is taken seems to have no influence. I cannot see at a glance why he has the right to do it! So the question is simply: WHY?