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How many possible Banach spaces are there on the entire set $\mathbb{R}$ ?

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How you mean how many? Are you interested in non isometrics one? Because there should be infinite many different banachspaces on $\mathbb{R}$ –  Dominic Michaelis May 9 '13 at 16:12
    
It depends on what you mean by different. For any $c>0$, the norm $\|x\|_c = c |x|$ is a different norm, and $(\mathbb{R}, \|\cdot \|_c)$ is a Banach space. –  copper.hat May 9 '13 at 16:13
    
You don't really need the emphasis that "on the entire set ${\mathbb R}$" provides. Since a Banach space is also a vector space (over ${\mathbb R},$ in this case), the question obviously doesn't arise for a proper subset of ${\mathbb R}.$ –  Dave L. Renfro May 9 '13 at 19:17
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Assuming you meant upto isomorphism, there is exactly one, since all norms on $\mathbb{R}$ are equivalent. ( in fact, any norm on $\mathbb{R}$ looks like $c|x|$) (Your definition of isomorphism doesn't matter in this case)

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