Why is $||\cdot||_1=\sum_{i=1}^n |x_i|$ not strictly subadditive (in $\mathbb{R}^2$)?

Question

Why is $||x||_1=\sum_{i=1}^n |x_i|$ not strictly subadditive (in $\mathbb{R}^2$)?

Intuitively one would start considering

$$||x||_1 +||y||_1$$

and want to display that this is always at least sometimes equal to $||x+y||_1$ I think.

One can write

$$\sum_{i=1}^n |x_i| + \sum_{i=1}^n |y_i|$$

Next I would probably combine the terms like

$$|x_1|+|y_1|+...+|x_n|+|y_n|$$

which is

$$\sum_{i=1}^n (|x_i|+|y_i|) \not=||x+y||_1$$

Then what?

Answer 1

No, a proof would be supplying an example for which one has equality in the inequality. Just consider the the vectors $x = (1, 1)$ and $y = (1,1)$ for example (there are more trivial examples). Then you can calculate $$ \Vert x + y \Vert_1 = 4 = 2 + 2 = \Vert x \Vert_1 + \Vert y \Vert_1.$$ Hence the inequality isn't strict.

Answer 2

In all generality, if you have a norm $||\ .||$ on a (real) vector space and take a vector $v$ and $\lambda>0$, you always have $||v+\lambda v || = ||v||+\lambda||v||$. In other words, norms are never strictly subadditive.