The question is decide whether or not $x_1^2 y_1^2 + x_2^2 y_2^2 + x_3^2 y_3^2$ in $R^3$ is an inner product.
I though the answer was yes, but the text answer sheet says no, and I can't figure out why.
I think my mistake is when trying to check $<x,y+w>$ or $c<x,y>$ for this.
I want to say that
$<x,y+w>=x _1^2 [y_1^2 + w_1^2] + x_2^2 [y_2^2 + w_2^2] + x_3^2 [y_3^2 + w_3^2]$
$=x _1^2 y_1^2 +x _1^2 w_1^2 + x_2^2 y_2^2 + x_2^2 w_2^2 + x_3^2 y_3^2 + x_3^2 w_3^2$
$=x _1^2 y_1^2+x_2^2 y_2^2+x_3^2 y_3^2+ x _1^2 w_1^2+x_2^2 w_2^2+x_3^2 w_3^2$
$=<x,y>+<x,w>$
For a scalar 'c' in R, $c<x,y>=<cx,y>$.
I want to say it is this, but not sure:
$c(x_1^2 y_1^2 + x_2^2 y_2^2 + x_3^2 y_3^2)$
$=c(x_1^2) y_1^2 + c(x_2^2) y_2^2 + (cx_3^2) y_3^2$
$=<cx,y>$
I am sure the other axioms hold.
$<x,y>=<y,x>$
and of course they are squared so $<x,x>$ is greater than or equal to zero (equal if and only if x = 0)
Thanks for taking a look.
$$ \langle \lambda x,y \rangle \neq \lambda \langle x,y \rangle $$ for so many choices of $\lambda \in \mathbb{R}$, $x$ and $y$; actually, the left-hand side is quadratic in $\lambda$, while the right-hand side is linear. Hence your quantity cannot be an inner product.