I would like to prove the following:

There is no linear isometry integrating $ ( mathbb {R} ^ 2, | cdot | _2) $ at $ (l ^ 1, | cdot | _1) $

I do not know how to prove it. Until now, I can only prove this result in the case where the vector $ (1,0) $ and $ (0.1) $ are sent to sequences that all have a positive value or a negative value.

In this case, I use the fact that the $ 2 $ the norm is not linear whereas the $ 1 $ the norm is (i.e. $ | xa + yb | = xa + yb $, $ x, y, a, b> $ 0).

The problem is that when the sequences have different signs, it's difficult to conclude.

Thank you.