 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2011/4/b/iv/solution

Since every

e_{n}

is nonzero, boundedness gives

∣ α_{n} ∣ ∥ e_{n} ∥_{\infty} = ∥ T e_{n} ∥_{\infty} \leq ∥ T ∥ ∥ e_{n} ∥_{\infty} .

Cancel its norm to obtain

n \in Z sup ∣ α_{n} ∣ \leq ∥ T ∥

. Thus the multiplier sequence is bounded.

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