Question

Prove there exists a positive number c such that two inner products with corresponding norms?

Answer 1

Answer:

See explanation below

Step-by-step explanation:

We assume that we have two inner products $_1,_2$ on v such that $=0$ if and only if $_2 =0$ and we want to proof is there is a positive number c like this :

$_1 \leq c_2$ for every $v,w$ in V

We can assumee that we have an orthonormal basis ${v_1,....,v_n}$ of V with respect $_1$. We can define the following sets:

$x= (x_1, x_2,...,x_n), y=(y_1 ,....,y_n)$ both defined on a n dimensional space, and then we have that for any linear comibnation we have this:

$_2 = x'Ax$

And A neds to be a matrix with entries $a_{ij}=$

So then we have this:

$r(x) = \frac{||x||_2}{||x||_1} =\frac{x' A x}{x' x}$

And then we have the maximum defined and we need to satisfy that c is the maximum value for te condition required.

$_1 \leq c_2$ for every $v,w$ in V