Dense Subspaces of Sobolev Spaces

Hello!

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?