In Evan’s book,

where

In Hunter’s *Applied Analysis, consists of functions that are uniformly continuous in .*

ect. (At least four different definitions appear as I asked in the previous question.)

*My question is, do definitions of ( , ) completely depend on context (so that there is no “the” definition)? Do they have any “stronger”, “weaker” relationship to each other or are these definitions not comparable at all?*

and

In Hunter’s Applied Analysis, consists of functions whose partial derivatives of order less than or equal to are uniformly continuous in .

Also, I’ve seen some people define as functions in that can be extended as functions in .

Leoni points out in his First Course in Sobolev Spaces that while the definitions of the spaces and $C^\infty(U)$ are standard, in the literature there are different definitions of the spaces and , and “unfortunately these definitions do not coincide”.

I’ve seen defined as in an undergraduate real analysis course that

Would you clarify what is really going on with the spaces ? How do PDE people usually think about such issue when a general open domain is considered?

*[Sorry, I do not really understand the question here. As Leoni states, there is no consensus on the notation here: each paper or text simply uses the notations that are most convenient for the application at hand, and one has to read the notational conventions carefully in those cases. It’s also worth bearing in mind that notation is ultimately an artificial human invention, rather than an innate feature of the mathematics one is working on; sometimes, two writers happen to use the same symbol to denote two rather different concepts, but this does not necessarily mean that these concepts have any deeper connection to them. -T.]*