Theorem 2.5.1 (Riesz-Thorin).label Let $1 \le p_{0}, p_{q}, q_{0}, q_{1} \le \infty$ and $\theta \in (0, 1)$. Let $1 \le p, q \le \infty$ with
\[\frac{1}{p}= \frac{1 - \theta}{p_{0}}+ \frac{\theta}{p_{1}}\quad \frac{1}{q}= \frac{1 - \theta}{q_{0}}+ \frac{\theta}{q_{1}}\]
Let $T: (L^{p_0}+ L^{q_0}) \to(L^{p_1}+ L^{q_1})$ with
\[M_{0} = \norm{T}_{L(L^{p_0}; L^{q_0})}\quad M_{1} = \norm{T}_{L(L^{p_1}; L^{q_1})}\]
then $T$ extends uniquely as a bounded map $L^{p} \to L^{q}$ with
\[\norm{T}_{L(L^p; L^q)}\le M_{0}^{1 - \theta}M_{1}^{\theta}\]