10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
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Quotients<br />
Lemma<br />
Let X be a <strong>Banach</strong> space and J ⊆ X a closed subspace. Then X/J<br />
is a <strong>Banach</strong> space in the quotient norm.<br />
Proof.<br />
Let {yn} ∞<br />
n=1 be a Cauchy sequence in X/J. For each n there is an<br />
Nn ∈ N such that yi − yj < 2−n−1 when i,j ≥ Nn. We arrange<br />
that N1 < N2 < N3 < ....<br />
Set zj = yNj and note that zj − zj+1 < 2−j for all j = 1,2,3,... .<br />
and note that<br />
Choose any x ′ j ∈ X such that q(x′ j ) = yj. Set x1 = x ′ 1<br />
x1 − x ′ 2 + J < 2−1 . It follows that there is a j2 ∈ J such that<br />
x1 − x ′ 2 − j2 < 2 −1 . Set x2 = x ′ 2 + j2 and note that q (x2) = z2<br />
and that x1 − x2 < 2 −1 .<br />
Note that x2 − x ′ 3 + J < 2−2 . It follows that there is a j3 ∈ J<br />
such that x2 − x ′ 3 − j3 < 2 −2 . Set x3 = x ′ 3 + j3 and note that<br />
q (x3) = z3 and that x2 − x3 < 2 −2 .<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>