Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 23H / a

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- Solution a

Solution

Strong convergence means

∥ f_{j} - f ∥_{p} \to 0

; weak convergence means

\int f_{j} h \to \int f h

for every

h \in L^{q}

. By duality choose

h

of norm one with

\int f h

arbitrarily close to

∥ f ∥_{p}

. Holder inequality and passage to the limit give

∥ f ∥_{p} \leq lim inf ∥ f_{j} ∥_{p}

.

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