What is "approximation in a Sobolev Space"? For example,

Question

I want to know the meaning of the statement as below.

$$ \text{There is a sequence} \;\{f_k\} \subset W^{3,2}\; \text{approximating}\;\; f \;\;\text{in}\;\; W^{2,2}. $$

Here $ W^{n,m} $ means a Sobolev Space.

Answer 1

Presumably this means that each $f_k$ is in $W^{3,2}$ and $f_k\rightarrow f$ in the norm of $W^{2,2}$ (i.e. $\|f_k-f\|_{W^{2.2}}\rightarrow 0$).