Is it possible to define a norm ||•|| on R^2 such that (1,0),(0,1) have norm 1 and ||(1,1)||<1?

Question

Is it possible to define a norm ||•|| on R^2 such that (1,0),(0,1) have norm 1 and ||(1,1)||<1? I don’t come up with any idea. I don’t know this assertion is correct or not.

Answer 1

Take $\|(x,y)\|=\frac 3 4 |x-y|+\frac 1 4 |x+y|$.

Answer 2

$\|(x,y)\|=\sqrt{x^2-(7/4)xy+y^2}$ should do.