Lemma 18.3.3.label Let $\dpn{E, F}{\lambda}$ be a duality over $\real$ and $f: E \to (-\infty, \infty]$, then the following are equivalent:
- (1)
$f^{*} \ne \infty$.
- (2)
There exists $(\phi, \alpha) \in F \times \real$ with $(\phi, \alpha) \le f$.
Proof. (1) $\Rightarrow$ (2): Let $\phi \in \bracsn{f^* < \infty}$, then by Definition 18.3.2, $(\phi, f^{*}(\phi)) \le f$.
(2) $\Rightarrow$ (1): By Definition 18.3.2, $f^{*}(\phi) \le \alpha$.$\square$
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