Consider in [X.sub.m]([[DELTA].sub.n]) the

seminorm [mathematical expression not reproducible].

We can show the convergence in the

seminorm [P.sub.[lambda],k] as follows.

In this note, we say that a subset X [subset] [W.sup.n,1.sub.loc] is bounded, if X is bounded for every

seminorm [[parallel] * [parallel].sup.T.sub.n,1], T [greater than or equal to] 0.

Of course, this value defines a

seminorm on [AL.sub.p,[eta]].

with the broken gradient [[nabla].sub.h] and the jump

seminorm [[absolute value of x].sub.j] yet to be designed.

Then for each continuous

seminorm p on E there exists an ([x.sub.0], [y.sub.0]) [member of] Gr(F) such that

For simplicity of presentation, we will use [W.sup.m,p] (E) to denote the usual Sobolev spaces that provided the norm [[parallel]*[parallel].sub.m,p,E] and the

seminorm [[absolute value of x].sub.m,p,E] for any 2D domain E.

If P = 2, we denote the Sobolev space by [H.sup.m] (G) and use the standard abbreviations [[parallel]*[parallel].sub.m,G] and [[absolute value of (*)].sub.m,G] for the norm and

seminorm, respectively.

(ii) Y is a

seminorm; that is, Y([sigma][M.sub.k]) = [absolute value of [sigma]]Y([M.sub.k]) and Y([M.sub.k1] + [M.sub.k2]) [less than or equal to] Y([M.sub.k1]) + Y([M.sub.k2]), where [M.sub.k1], [M.sub.k2] [member of] M, and [sigma] [member of] R.

The symbol [[integral].sub.X] [absolute value of ([nabla][[mu].sub.a])] denotes the total variation

seminorm [32] of [[mu].sub.a] [member of] [L.sup.1] (X) as follows:

It is a familiar property for the Frechet module E that a

seminorm [sigma] on E is [T.sub.E]-continuous, if and only if there is a

seminorm [p.sub.i] [member of] P and a positive finite constant [C.sub.i] such that

Assume that, for any open subsets [O.sub.1], [O.sub.2] of X such that [O.sub.1] [subset] [O.sub.2], we have I([O.sub.1]) [subset] I([O.sub.2]) and if [[rho].sup.2.sub.1] is the restriction operator E([O.sub.2]) [right arrow] E([O.sub.1]), then, for each [p.sub.i] [member of] P([O.sub.1]), the

seminorm [[??].sub.i] = [p.sub.i] [omicron] [[rho].sup.2.sub.1] extends [p.sub.i] to P([O.sub.2]).