this is an Apostol problem in chapter 12 and I guess it's a hypothetical definition of a norm of a vector

Assuming this definition of the norm prove these statements-

do I just write out the sum components in the triangle equality proof? kind of odd since normally one defines norm as the square root of the dot product of a vector

Use this definition in V2 and prove on a figure the set of all points (x,y) of norm 1- this is just the line x+y=1?

which of the above theorems/statements would hold if we defined the norm as an absolute value?

anyways I feel like I don't have a lot of confidence from this forum!