### Some Facts from Functional Analysis

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As is well known and easily veried,

4 d

0 2s converges when 2s 1, and thus when s 1 . We must now estimate 1 d

2

Since cos 0 for [ , ], (1 r2 2r cos )s . 1 r2 2r cos 1 r2

for such . Therefore 1 d 1 d 1 1 (1 r2 2r cos )s
(1 r2)s 2(1 r2)s 2 . Thus, for every r [0, 1), M (r) 1

2 d

2s 4 d 1 sin2s
2s 2 . Since this bound on M (r) is independent of r, it follows from Theorem that f is in H2.
Another space of analytic functions arises in the study of operators on H2. Definition . The space H consists of all the functions that are ana- lytic and bounded on the open unit disk. The...

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