 Solution

ID: past-exam-of-the-options-ii-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/d4/e/i/solution

For each integer

m \geq 1

, list the

m^{2}

f_{m, j} (x) = m 1_{[j / m^{2}, (j + 1) / m^{2})} (x), 0 \leq j < m^{2},

in increasing

j

, and concatenate the blocks in increasing

m

. Each function has

L^{1}

1/ m

, so the resulting sequence has norm tending to zero. Every

x \in (0, 1)

lies in exactly one interval of each block, where the function takes value

m

. Therefore

∥ f_{n} ∥_{1} \to 0, n lim sup f_{n} (x) = \infty (0 < x < 1) .

This unbounded typewriter sequence with vanishing L1 norm is a moving-spike construction: the support shrinks fast enough to beat the increasing height.

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