 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2011/5/a/ii/solution

Put

w = x - y_{0}

. For any

y \in Y

and any real

t

, minimality gives

∥ w - t y ∥^{2} - ∥ w ∥^{2} = - 2 t Re ⟨ w, y ⟩ + t^{2} ∥ y ∥^{2} \geq 0.

The derivative at zero must vanish, so

Re ⟨ w, y ⟩ = 0

. In a complex Hilbert space, apply the same argument to

i y

to get the imaginary part zero as well. In the real case the first conclusion is already enough. Hence

x - y_{0} \in Y^{⊥} .

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