The example of a mapping is homomorphic but not isomorphic

Question

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.

Answer 1

Let $A=B=\Bbb R$ and $T(x)=2x$.