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I am looking for a textbook showing that (i) every compact operator is bounded and (ii) composing a compact operator with a bounded one (and a bounded operator with a compact one) gives a compact operator (in the setting of real or complex normed spaces). I am not interested in the proofs of these results. Instead, I am interested in a reference where these results are proved with no assumptions of completeness. This is not the case, for instance, of Rudin's Functional Analsysis (ed. 1991).

Thank you in advance for any help you can provide.

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    If your definition of "precompact" is my definition of "relatively compact" (cf. http://en.wikipedia.org/wiki/Precompact_set), then I agree with you: They are more or less immediate facts. Nonetheless, I need a reference: I am writing a paper, where these properties pop up while working out an example in a long list of examples ranging from graph theory to topology, from universal algebra to control theory. Thus, I'd like to be as exhaustive as possible, since the paper is not necessarily addressed to people with a (basic) background in functional analysis (and a reference doesn't cost much).2012-11-21

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