# real analysis – linear isometric integration of \$ ( mathbb {R} ^ 2, | | _2) \$ to \$ (l ^ 1, | | _1) \$

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.