In order to define a dot product, can we apply the polarization identity to any norm ?
One exercise I had was based on the norm 1 : $$||u||_1=\sum{|u_i|}$$ I wonder if I could have used (wasn't asked...) the polarization identity : $$<u,v>=\frac{1}{2}(||u+v||_1^2-||u||_1^2-||v||_1^2)$$ in order to define a "dot product 1" : $<u,v>_1$ ?
It doesn't pass the test of linearity.
To be an iner product space.
$\langle a\mathbf{u,v}\rangle = a\langle\bf{ u,v}\rangle$
let $\mathbf u = (1,1), \mathbf v = (1,-1)$
$\|\mathbf u\|_1 = 2, \|\mathbf v\|_1 = 2, \|\mathbf{u+v}\|_1 = 2,\frac 12(2^2 - 2^2 - 2^2) = -2$
$\|\mathbf 2u\|_1 = 4, \|\mathbf v\|_1 = 2, \|\mathbf{u+v}\|_1 = 4,\frac 12(4^2 - 4^2 - 2^2) = -2$