The analytic function theory is the most powerful tool in the operator theory. Here we briefly recall few facts of complex analysis used in this course. Use any decent textbook on complex variables for a concise exposition. The only difference with our version that we consider function f(z) of a complex variable z taking value in an arbitrary normed space V over the field ℂ. By the direct inspection we could check that all standard proofs of the listed results work as well in this more general case.
| f′(z0)= |
|
| . (108) |
| f(z)= |
| ck zk, for some ck∈ V. (109) |
| r′= |
| ⎪⎪ ⎪⎪ | cn | ⎪⎪ ⎪⎪ | 1/n and |
| = |
| ⎪⎪ ⎪⎪ | cn | ⎪⎪ ⎪⎪ | 1/n. (110) |
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