We can define a 'distance' between two points $P = (x_1,y_1)$ and $Q=(x_2,y_2)$ of the plane by $d(P,Q) = |x_2 – x_1| + |y_2 – y_1|$. Verify if the sentence below is a inner product in the plane.
$$\langle(x_1,y_1),(x_2,y_2)\rangle = d(P,Q)$$
UPDATE: What i've already made
Positivity
$\langle P,P\rangle \geq 0 $
The distance from a point to a point is 0. [check]
Symmetry
$ \langle P,Q \rangle = \langle Q,P \rangle$
The distance from a point P to a point Q is equal to distance from a point Q to the point Q.[check]
Bilinearity
$ \langle\lambda P,Q\rangle = \lambda\langle P,Q\rangle \\
\langle(\lambda x_1,\lambda y_1),(x_2,y_2)\rangle = |x_2 – \lambda x_1| + | y_2 – \lambda y_1 |\\$
I'm stuck here, i don't know how get out with this.
Best Answer
It's not an inner product because it isn't linear. For example, the choice $P=Q=(1,\,0)$ implies $d(P,\,kQ)=|k-1|$, which for $k\ne 1$ differs from $kd(P,\,Q)=0$.