 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2010/a/1/d/solution

(α, β) = (0, e_{n})

. Its input norm is one, while

T_{1} (0, e_{n}) = (e_{n}, 0)

has norm

n^{2}

. Thus $T_{1}$ is unbounded.

For the second map, since

j^{2} \geq 1

∥ T_{2} (α, β) ∥_{X} = ∥ α + β ∥_{1} \leq ∥ α ∥_{1} + ∥ β ∥_{1} \leq ∥ α ∥_{1, 2} + ∥ β ∥_{1} = ∥ (α, β) ∥_{X} .

Equality holds at

(0, e_{1})

, so

T_{2} is bounded and ∥ T_{2} ∥ = 1.

c_{00} \times c_{00}

into itself because adding two finitely supported sequences preserves finite support.

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