 Solution

ID: past-exam-of-the-functional-analysis-and-banach-and-c-algebras-course-of-the-university-of-oxford/2007/a/4/iv/solution

Suppose

T_{n}

are compact and

∥ T_{n} - T ∥ \to 0

. For any

ε > 0

take

n

with

∥ T_{n} - T ∥ < ε

. A finite

ε

-net of

T_{n}

of the unit ball is then a finite

2 ε

-net of its

T

T

is compact by the stated criterion. Hence $K (X)$ is norm closed in $B (X)$ .

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