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Functional analysis:)

A pair of

real numbers p,q>1 satisfying 1/p+1/q=1.

Without motivation, this is completely idiotic. So here's some...

A version of Hölder's inequality with integrals immediately bounds the result of "applying" a function in L_{q} to a function in L_{p}. So any function in L_{q} is a linear functional on L_{p}: in symbols, L_{q}⊆(L_{p})^{*}. In fact, equality holds: L_{q} is the space of linear functionals on L_{p} (and, by symmetry, L_{p} is the space of linear functionals on L_{q}).

For p=1, we "have" q=∞. Except that this is of course not true, and the preceeding paragraph breaks down. It is still true that (L_{1})^{*}=L_{∞}, but the proof is somewhat different. And L_{∞} is a completely different sort of space: typically, (L_{∞})^{*}**!=**L_{1}, but is much larger.