Note 3
Compact Operator
1
Sobolev space H n
Sobolev norm ||| f |||2n : = ∑|α|≤n ||Dαf||22
||| f ||||n :=
(∑|α|≤n|yα|2Ff(y)|2dNy
(∑|α|≤n|yα|2Ff(y)|2dNy
||| f ||||n < ∞
f ∈ L2 ∈ H n
H n is Hilbert space by inner product corresponded with norm ||| ・ |||.
2
Operator in Hilbert space H A
Unit sphere of Sobolev space H B
Compact subset of H
Complete orthonormal system of H {φn }∞n=1
Pnf : = ∑n r=1 <f, φr >φr
Pn is finite class operator.
1-Pn is convergent over D by the next.
D is all bounded.
Arbitrary ε> 0
Finite set of D {x1, …, xs}
x ∈ D ||x – xt || < ε/ 2 1 ≤ t ≤ s
N ≤ n and x ∈ D
||(1-Pn)x|| < ||(1-Pn)xt|| + ||(1-Pn)(xt-x)|| ≤ε/ 2 +ε/ 2 =ε
3
A is compact operator.
[References]
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