Remark 5.17.6. Here are some references to the literature. In [I, p. 75, Theorem 1, Bourbaki] you can find: (2) $\Leftrightarrow $ (4). In [I, p. 77, Proposition 6, Bourbaki] you can find: (2) $\Rightarrow $ (1). Of course, trivially we have (1) $\Rightarrow $ (4). Thus (1), (2) and (4) are equivalent. The equivalence of (3) and (4) is [Chapter II, Exercise 25, LangReal].
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