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.