10. Commutative Banach algebras - Aarhus Universitet

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Quotients Proof. We continue by induction to construct a sequence {xj} ∞ j=1 such that a) q (xj) = zj = yNj , b) xj − xj+1 < 2 −j , in X for all j. It follows from b) that xm − xn ≤ xm − xm−1 + xm−1 − xm−2 + · · · + xn+1 − xn ≤ m−1 j=n 2−j . Thus {xj} is a Cauchy sequence in X and x = limj→∞ xj exists in X because X is complete. We finish the proof by checking that y = q(x) is the limit {yn} in X/J: Klaus Thomsen 10. Commutative Banach algebras
Quotients Proof. First we observe that y − zj = q(x) − q(xj) ≤ x − xj, proving that that y = limj→∞ zj in X/J. Let ǫ > 0. Since {yn} is Cauchy there is an M ∈ N such that yn − ym < ǫ when n,m ≥ M. In particular, zj − ym < ǫ for all m > M and all sufficiently large j. By letting j tend to infinity we deduce that y − ym < ǫ for all m > M. Klaus Thomsen 10. Commutative Banach algebras
Page 1 and 2: 10. Commutative Banach algebras Kla
Page 3 and 4: The Gelfand-Mazur theorem Let A and
Page 5 and 6: Commutative Banach algebras - ideal
Page 7 and 8: The proof of Theorem 11.3 Proof. (b
Page 9: Quotients Lemma Let X be a Banach s
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
Page 23: The Gelfand transform - the proof o

10. Commutative Banach algebras - Aarhus Universitet

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