and equipped with the norm
and the scalar product ,
and the space
Next, we define the space with
where and Einstein's summation convention is applied to indices and , and
It is a standard result that is a norm of . The following lemma (cf. ) allows us to represent the norm of the .
Lemma 1 For all the equationProof. Assume . We rewrite the operator as follows
and after having applied partial integration twice we arrive at
Let us discuss further spaces, cf. . We define
It is proved in  that
We remark that has a zero trace of the normal component on , whereas has a zero trace of the tangential component. Therefore, the space seems to be appropriate to the flux line boundary condition, cf. (12). Since we are interested in formulating a gauge condition that involves the divergence of the vector potential, we introduce the space
7], we have the following results.
Lemma 2 Assume that is bounded, Lipschitz-continuous, simply-connected and has just one component. Then there exists a positive constant such thatProof. See , Lemma 3.4.
Lemma 3 Assume that is bounded, Lipschitz-continuous, simply-connected and has just one component. Then, the seminormProof. Lemma 3 is a direct consequence of Lemma 2.
The next lemma requires more than the standard assumptions for technical magnetic field problems. Indeed, electromagnetic devices may have re-entrant corners. Therefore, we will not base our further investigations on Lemma 4.
Lemma 4 Assume further that either has a boundary, or is a convex polyhedron. Then is continuously imbedded in .Proof. See , Theorem 3.7.