 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2010/b/5/i/solution

T = T^{*}

, equivalently

⟨ T x, y ⟩ = ⟨ x, T y ⟩

for every

x, y

T x = 0

, then

⟨ x, T y ⟩ = ⟨ T x, y ⟩ = 0

for all

y

, so

x \in (Ran T)^{⊥}

. Conversely, membership in that orthogonal complement gives

⟨ T x, y ⟩ = 0

for every

y

, and choosing

y = T x

yields

T x = 0

. Therefore

ker T = (Ran T)^{⊥} .

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