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I have studied that every normed space $X$ is homeomorphic to its open unit ball $B$. I want to know what conclusion can we draw from this statement? Does it mean that every normed space is open ball? I am confused with this statement. Could anyone clear my doubt?

Thanks for help

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up vote 5 down vote accepted

It is true that every normed linear space $X$ is homeomorphic to its open unit ball $B$, for example via the homeomorphism $f:B\to X$ defined by $f(x)=\tan(\frac{\pi}{2}\|x\|)\cdot x$. This means that, as topological spaces, $X$ and $B$ are essentially identical. However, a normed linear space carries more information than a topological space, namely a norm. Thus the appropriate notion of "essentially identical" for normed linear space is the existence of an isometric isomorphism, a homeomorphism which preserves the norm. Clearly $X$ is not isometrically isomorphic to $B$, since $B$ is not even a normed linear space unless $X=\{0\}$. To see this, take any $x\neq 0$. Then $x/2\|x\|\in B$ but $2\cdot x/2\|x\|\notin B$.

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Thank you very much sir. – srijan Jul 4 '12 at 5:24
As a normed linear space, $B$ isn't. – Robert Israel Jul 4 '12 at 5:43
@RobertIsrael Indeed, I should probably mention that. – Alex Becker Jul 4 '12 at 5:51
$B$ can't have a norm because it's not a linear space, but it does have a metric. $X$ is not isometric to $B$ ($B$ is bounded, but $X$ is not). – Robert Israel Jul 4 '12 at 17:02

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