I know that direct limit always exist. But if we follow the same proof, constructing the direct product, how do we know the norm is finite？
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Your title seems to contradict the first statement made in your question. Is your question how do we know that the norm is finite, or whether or not direct limits always exist? – Aweygan Jul 10 '18 at 19:10
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I mean direct limit always exist, how about inverse limit？ – user540663 Jul 11 '18 at 06:06
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Oh yes, I'm sorry, I thought you were only referring to direct limit. – Aweygan Jul 11 '18 at 11:57