Homework 11, due Friday 7 April.
Given a compact subset \( K \) of the complex plane, let \( R(K) \) denote the rational functions with poles off \( K \). Suppose \( \mathscr{A} \) is a unital commutative Banach algebra. Show, if \( a\in \mathscr{A} \), then there is a natural unital homomorphism \( R(\sigma(a))\to \mathscr{A}, \, r\mapsto r(a) \) sending the identity function to a. Show, if the set \( \{r(a):r\in R(\sigma(a))\} \) is dense in \(\mathscr{A} \), then the maximal ideal space of \( \mathscr{A} \) is homeomorphic to the spectrum of a.
Homework 10, due Friday 24 March. Problem 25.8. (17 Mar)
Homework 9, due Friday 17 March. Problem 25.4. (26 Feb)
Homework 8, due Friday 24 February. Problem 24.7, with \( 1\le p,q <\infty.\) (17 Feb)
Homework 7, due Friday 17 February. (10 Feb). Suppose \( H \) is a Hilbert space, \( (h_n) \) is a sequence from \( H \) and \( h\in H. \)
Prove \( (h_n) \) converges to \( h \) in norm if and only if (i) \( (h_n)\) converges to \( h \) weakly and (ii) \( \|h_n\|\to \|h\|. \)
Prove if \( (h_n) \) converges to \( h \) weakly, then \( \|h\|\le \liminf \|h_n\|. \)
Homework 6, due Friday 10 February. Problem 23.2. (3 Feb).
Homework 5, due Friday 3 February. Use Corollary 23.17 to do Problem 23.5. (27 Jan).
Homework 4, due Friday 27 January. Problem 22.2 (be sure to use the problem statement from the notes dated 20jan2017 or later). (20 jan).
Homework 3, due Friday 20 January. Problem 22.6 (13 jan).
Homework 2, due Friday 13 January. Problem 21.7 (9 jan)
Homework 1, due Wednesday 11 January. Problem 21.3 (6 jan).
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