 Solution

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

Use the convention that the inner product is linear in its first argument. By the defining identity for the adjoint operator,

x \in (Im T^{*})^{⊥} ⟺ ⟨ x, T^{*} y ⟩ = 0 for every y \in X ⟺ ⟨ T x, y ⟩ = 0 for every y \in X ⟺ T x = 0.

For the last implication choose

y = T x

Im T^{*}

gives

ker T = (Im T^{*})^{⊥}, (ker T)^{⊥} = \overline{Im T^{*}} .

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