The example of two Banach space A,B such that the mapping T:A→B, is onto , one-to-one and homomorphic but not isomorphic i.e ∥T(x)∥≠∥x∥. I think there are two norm spaces, such that norm does not lead to the inner product.
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You mean isometric, not isomorphic I assume, since the open-mapping theorem implies that the inverse is continuous. – Adam Hughes Nov 19 '14 at 07:50
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1Let $A=B=\Bbb R$ and $T(x)=2x$. – Adam Hughes Nov 19 '14 at 07:51
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@AdamHughes: You should write that as an answer. – Michael Albanese Dec 16 '14 at 20:00
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1@Michael Albanese Okie dokie. – Adam Hughes Dec 16 '14 at 20:35
