 Solution

ID: past-exam-of-the-advanced-topics-in-statistical-machine-learning-course-of-the-university-of-oxford/2025/2/b/i/solution

Take the real L2 space

H = L^{2} ((0, \infty), d t)

φ (x) (t) = 1_{(0, x]} (t) .

H

because its squared L2 norm is

x < \infty

. Their inner product is the length of the overlapping intervals:

⟨ φ (x), φ (x^{'})⟩ = \int_{0}^{\infty} 1_{(0, x]} (t) 1_{(0, x^{'}]} (t) d t = min (x, x^{'}) = k_{1} (x, x^{'}) .

This realizes

k_{1}

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