Lemma 28.2.5 (Hadamard’s Three Lines Lemma).label Let $S = \bracs{z \in \complex| \text{Re}(z) \in [0, 1]}$, $E$ be a Banach space over $\complex$, and $f \in H(S; E) \cap BC(\ol{S}; E)$. For each $s \in [0, 1]$, let
then for each $s \in [0, 1]$, $M(s) \le M(0)^{s} M(1)^{1-s}$.
Proof, [Theorem VI.3.7, Con78]. Assume without loss of generality that $M(0), M(1) > 0$. Let
then $g$ is a non-vanishing entire function, and for each $z \in \complex$,
so $|g|^{-1}$ is bounded on $\ol S$ by Proposition 5.16.2. Let
then $h \in H(S; E) \cap BC(\ol S; E)$ with $\norm{h(z)}_{E} \le 1$ for all $z \in \partial S$. By the Maximum Modulus Theorem, $\norm{h(z)}_{E} \le 1$ for all $z \in S$. Thus for every $z \in S$,
Therefore $M(s) \le M(0)^{s} M(1)^{1-s}$ for every $s \in [0, 1]$.$\square$
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