= Riesz-Thorin theorem
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{wiki=Riesz–Thorin_theorem}
= Riesz-Thorin interpolation theorem
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{synonym}
For a <linear operator> bounded from $L^{p_i}$ to $L^{q_i}$ with <norms> $M_i$, $i=0,1$, interpolate reciprocals by $1/p=(1-\theta)/p_0+\theta/p_1$ and $1/q=(1-\theta)/q_0+\theta/q_1$. The intermediate <operator norm> is at most $M_0^{1-\theta}M_1^\theta$. The complex interpolation proof first treats <simple functions>; completeness and <continuity> on the endpoint sum spaces identify the extension with the original operator. On finite counting spaces it also gives <norm> bounds between sequence spaces.
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