Answer:
See explanation below
Step-by-step explanation:
We assume that we have two inner products
on v such that
if and only if
and we want to proof is there is a positive number c like this :
for every
in V
We can assumee that we have an orthonormal basis
of V with respect
. We can define the following sets:
both defined on a n dimensional space, and then we have that for any linear comibnation we have this:
![_2 = x'Ax](https://tex.z-dn.net/?f=%20%3C%5Csum_%7Bi%3D1%7D%5En%20x_i%20v_i%2C%20%5Csum_%7Bk%3D1%7D%5En%20y_k%20v_k%3E_2%20%3D%20x%27Ax)
And A neds to be a matrix with entries ![a_{ij}=](https://tex.z-dn.net/?f=a_%7Bij%7D%3D%20%3Cv_i%2C%20v_j%3E)
So then we have this:
![r(x) = \frac{||x||_2}{||x||_1} =\frac{x' A x}{x' x}](https://tex.z-dn.net/?f=r%28x%29%20%3D%20%5Cfrac%7B%7C%7Cx%7C%7C_2%7D%7B%7C%7Cx%7C%7C_1%7D%20%3D%5Cfrac%7Bx%27%20A%20x%7D%7Bx%27%20x%7D)
And then we have the maximum defined and we need to satisfy that c is the maximum value for te condition required.
for every
in V