Apparently he is in a mental hospital at the very start of the book but i have read the whold book and did not realise this.
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FRom the text, we can infer that Holden is not in a regular hospital. He also uses the word "madman" and having to go somewhere to take it easy.
I'll just tell you about this madman stuff that happened to me around last Christmas just before I got pretty run-down and had to come out here and take it easy.
The Catcher in the Rye