Kenneth Kuttler

27.2. DUAL SPACES IN ORLITZ SPACE 991

Now by Theorem 11.3.9 applied to the positive and negative parts of real and imaginaryparts, there exists a uniformly bounded sequence of simple functions, {sk} converginguniformly to un, implying convergence in EA (Ω) , and so

Lun = limk→∞

Lsk = limk→∞

∫Ω

skvdµ =∫

Ω

unvdµ. (27.2.24)

therefore, from 27.2.23,

||un||A ≤r||L||

∫Ω

unvdµ =r||L||

L(un)≤r||L||||L|| ||un||A ,

which is a contradiction since r < 1. Therefore, ||un||A ≤ 1 and from 27.2.23,

||L|| ≥ ||L|| ||un||A ≥ |Lun|=∫

Ω

unvdµ

≥ ||L||∫

Ã(

r |v(x)|||L||

)dµ.

Letting n→ ∞ the monotone convergence theorem and the above imply∫Ω

Ã(

r |v(x)|||L||

)dµ ≤ 1

which shows that v ∈ LÃ (Ω) and ||v||Ã ≤||L||

r for all r ∈ (0,1) . Therefore, ||v||Ã ≤ ||L|| .Since v ∈ LÃ (Ω) it follows Lv = L on S and so Lv = L because S is dense in the set

EA (Ω) . The last assertion follows from Proposition 27.1.10. This completes the proof.

27.2. DUAL SPACES IN ORLITZ SPACE991Now by Theorem 11.3.9 applied to the positive and negative parts of real and imaginaryparts, there exists a uniformly bounded sequence of simple functions, {s;,} converginguniformly to u,, implying convergence in E, (Q), and soLu, = jim n Ls = = lim ff savd Lb = L Unvd LL.therefore, from 27.2.23,r< — d -~ 7, Lwhich is a contradiction since r < 1. Therefore, ||u,||, <1 and from 27.2.23,IIL 1|IV| lunlla > Heel = ff waveyell fa) ay,Letting n — oo the monotone convergence theorem and the above implyLa2)or=(27.2.24)which shows that v € Lj (Q) and ||v||z < et for all r € (0,1). Therefore, ||v||7 < ||L]|.Since v € Lz (Q) it follows L, = L on S and so Ly = L because S is dense in the setE, (Q). The last assertion follows from Proposition 27.1.10. This completes the proof.