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I am reading Functional analysis by M. Fabian. Author gave the proof that C[0,1] is Banach space and then says that we can generalise the result for any compact set K. And now after reading this generalise statement I am wondering where he has actually use the closeness and boundedness of [0,1]. Like if I take (0,1) where will proof go wrong?( The proof I guess will be similar in all texts).

José Carlos Santos
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ogirkar
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  • https://www.google.com/url?sa=t&source=web&rct=j&url=https://math.stackexchange.com/questions/83830/how-to-show-that-c-c0-1-is-a-banach-space&ved=2ahUKEwiorb2Wu_vjAhWBNI8KHatWDy4QFjACegQICRAC&usg=AOvVaw2Q6T-miAnPG7Xwukjgslsl – ogirkar Aug 11 '19 at 18:37
  • Same proof as above answer – ogirkar Aug 11 '19 at 18:38
  • Banach with respect to the supremum norm? Then for $(0,1)$ this is not a norm – G. Chiusole Aug 11 '19 at 18:40

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In $[0,1]$ (or in a general compact space $K$), the distance that we are using is$$d(f,g)=\sup_{x\in[0,1]}\bigl\lvert f(x)-g(x)\bigr\rvert.$$In, say, $(0,1)$ this metric doesn't make sense, because then there are unbounded continuous functions and therefore that $\sup$ may not exist.

David C. Ullrich
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José Carlos Santos
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