efinition
(Sobolev spaces
with dominating mixed derivative) For a domain
we introduce the
classes
See the section (
Tensor
product of Hilbert spaces
) for definition of tensor product of Hilbert
spaces and associated scalar product.
The above definition is to be compared with the following statement.
Proposition
(Sobolev spaces in N
dim as tensor products) For a domain
we have
Proof
The spaces on the left and right of
have equivalent norms, the intersection of sets corresponds to summation of
norms. They also coincide as sets. So see this it suffices to use the
proposition (
Tensor product of
function spaces
) to construct bases for dense subsets.
