10. Commutative Banach algebras - Aarhus Universitet

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Quotients Let X be a Banach space and J ⊆ X a closed subspace. Then the quotient space X/J comes equipped with the norm x + J = inf {x − j : j ∈ J} , called the quotient norm. The reader should check that this is indeed a norm on the quotient space. E.g the triangle inequality is established as follows: When x,y ∈ X and j,j ′ ∈ J, x + y + J ≤ x + y − j − j ′ ≤ x − j + y − j ′ . By taking the infimum over all j ′ ∈ J we conclude that x + y + J ≤ x − j + y + J; then we take the infimum over all j ∈ J to conclude that x + y + J ≤ x + J + y + J. In the following we will often write q(x) = x + J and consider q : X → X/J as a linear surjection - the quotient map. Note that q(x) ≤ x for all x ∈ X. Klaus Thomsen 10. Commutative Banach algebras
Quotients Lemma Let X be a Banach space and J ⊆ X a closed subspace. Then X/J is a Banach space in the quotient norm. Proof. Let {yn} ∞ n=1 be a Cauchy sequence in X/J. For each n there is an Nn ∈ N such that yi − yj < 2−n−1 when i,j ≥ Nn. We arrange that N1 < N2 < N3 < .... Set zj = yNj and note that zj − zj+1 < 2−j for all j = 1,2,3,... . and note that Choose any x ′ j ∈ X such that q(x′ j ) = yj. Set x1 = x ′ 1 x1 − x ′ 2 + J < 2−1 . It follows that there is a j2 ∈ J such that x1 − x ′ 2 − j2 < 2 −1 . Set x2 = x ′ 2 + j2 and note that q (x2) = z2 and that x1 − x2 < 2 −1 . Note that x2 − x ′ 3 + J < 2−2 . It follows that there is a j3 ∈ J such that x2 − x ′ 3 − j3 < 2 −2 . Set x3 = x ′ 3 + j3 and note that q (x3) = z3 and that x2 − x3 < 2 −2 . Klaus Thomsen 10. Commutative Banach algebras
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Page 3 and 4: The Gelfand-Mazur theorem Let A and
Page 5 and 6: Commutative Banach algebras - ideal
Page 7: The proof of Theorem 11.3 Proof. (b
Page 11 and 12: Quotients Proof. First we observe t
Page 13 and 14: Theorem 11.5 Let A be a Banach alge
Page 15 and 16: Theorem 11.5 Proof. It follows that
Page 17 and 18: Theorem 11.5 Proof. (e) : If λ ∈
Page 19 and 20: The Gelfand transform - Theorem 11.
Page 21 and 22: The Gelfand transform - the proof o
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10. Commutative Banach algebras - Aarhus Universitet

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