 Solution

ID: past-exam-of-the-functional-analysis-and-banach-and-c-algebras-course-of-the-university-of-oxford/2009/a/3/iv/solution

(iv) implies (iii):

P_{k} = \sum_{n = 1}^{k} x_{n} φ_{n}

is a finite sum of bounded rank-one operators.

(iv) implies (v): continuity gives

φ_{n} (y_{r}) \to 0

whenever

y_{r} \to 0

, so any scalar limit

β

must vanish.

(iv) implies (vi): fix

n

. Continuity gives

φ_{n} (y_{r}) \to φ_{n} (x)

, while the assumed uniform coordinate convergence gives

φ_{n} (y_{r}) \to β_{n}

. Uniqueness of the scalar limit yields

β_{n} = φ_{n} (x)

for every

n

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