Respuesta :
Answer:
See explanation below
Step-by-step explanation:
We assume that we have two inner products [tex]<.,.>_1,<.,.>_2[/tex] on v such that [tex]<v,w_1>=0[/tex] if and only if [tex]<v,w>_2 =0[/tex] and we want to proof is there is a positive number c like this :
[tex]<v,w>_1 \leq c<v,w>_2[/tex] for every [tex]v,w[/tex] in V
We can assumee that we have an orthonormal basis [tex]{v_1,....,v_n}[/tex] of V with respect [tex]<,>_1[/tex]. We can define the following sets:
[tex] x= (x_1, x_2,...,x_n), y=(y_1 ,....,y_n)[/tex] both defined on a n dimensional space, and then we have that for any linear comibnation we have this:
[tex] <\sum_{i=1}^n x_i v_i, \sum_{k=1}^n y_k v_k>_2 = x'Ax[/tex]
And A neds to be a matrix with entries [tex]a_{ij}= <v_i, v_j>[/tex]
So then we have this:
[tex]r(x) = \frac{||x||_2}{||x||_1} =\frac{x' A x}{x' x}[/tex]
And then we have the maximum defined and we need to satisfy that c is the maximum value for te condition required.
[tex]<v,w>_1 \leq c<v,w>_2[/tex] for every [tex]v,w[/tex] in V
