ABSTRACT
F r o m this, for A belonging to V'L2 as an immedia te consequence of T h e o r e m 5.7.2 we ob ta in the fol lowing
THEOREM 7.5.3. If A is in T>'L2, then the function given by (7.5.3) fulfils the Laplace equation Au — 0 in I l + and l im y _^ 0 + ui'i y) = A in the sense of V'L2 (compare with Theorem 5.6.4)-
A P P E N D I X
Let I? be a vector space over the field of complex numbers. L e t us equip this space w i t h a fami ly of norms || • | | m fulfi l l ing the inequali t ies
IMIm < I M L + i for (f e E
and m = 0 ,1 , 2 . . . . A sequence (<pv) is called convergent to <p i n E i f for m G N , \\ipv — <p\\m tends
to zero as v oo. If for m G N , \\ipv — ^ | | m goes to zero as v and fx t end to infinity, then we say that the sequence (tpv) is a Cauchy sequence i n E.
