 Solution

ID: past-exam-of-the-functional-analysis-i-course-of-the-university-of-oxford/2023/3/c/i/solution

ℓ_{0}, ℓ_{1}

ℓ_{i} (e_{j}) = δ_{ij} (i, j \in {0, 1}) .

The endpoint values also make

e_{0}, e_{1}

T

lies in

E = ⟨ e_{0}, e_{1} ⟩

. Conversely

T e_{j} = e_{j}

, so

T

E

. Thus

Im T = ⟨ e_{0}, e_{1} ⟩, rank T = 2.

For every

f

T f \in E

, whence

T^{2} f = T f

. It is a linear projection, and an annihilating polynomial is

p (z) = z (z - 1), p (T) = T^{2} - T = 0.

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