 Solution

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

(v) implies (iv). Fix

n

and consider any sequence

y_{r} \to y

with

φ_{n} (y_{r}) \to γ

. Apply (v) to

y_{r} - y

and

β = γ - φ_{n} (y)

. It gives

γ = φ_{n} (y)

. Thus the graph of

φ_{n} : X \to K

is closed. Both spaces are Banach, so the closed graph theorem makes

φ_{n}

bounded.

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