 Solution

ID: past-exam-of-the-functional-analysis-and-banach-and-c-algebras-course-of-the-university-of-oxford/2008/a/2/ii/solution

The extension already agrees with the usual limit on

c

. Alternatively, applying the domination inequality to both

x

and

- x

gives

lim inf_{n} α_{n} \leq φ (x) \leq lim sup_{n} α_{n} .

When the sequence converges, both endpoints are its limit, proving $φ (x) = lim α_{n}$ . These bounds also give

∥ φ ∥ = 1

and positivity on nonnegative sequences, but do not by themselves assert shift invariance.

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