Let $V$ be a finite dimensional vector space over the field $\Bbb F$ ($\Bbb R$ or $\Bbb C$) and let $q: V\to\Bbb F$ be a mapping. Then is it true that the following two conditions are independent, that is one does not imply the other?
$q(ax)=a^2q(x)$ for all $a$ in $\Bbb F$ and for all $x$ in $V$.
The mapping defined by
$$\varphi_q(x , y)=q(x + y) - q(x) - q(y)$$
is a bilinear form on $V$.
As both the above conditions should be satisfied for the mapping $q: V\to\Bbb F$ to be called as a quadratic form on $V$. Hence my question is can there be a mapping $q: V\to\Bbb F$ such that condition (1) holds but (2) does not and vice-versa. Thanks for any help in advance.
The first condition only constrains the function values along each ray individually. For instance, if $V=\mathbb R^2$ over $\mathbb F=\mathbb R$, you could assign arbitrary values to $q$ on the unit circle and then extend $q$ to $V$ such that the first condition is fulfilled. Most such functions will not fulfil the second condition, which connects function values on different rays.
On the other hand, any linear function $q$ fulfils the second condition, since in this case $\phi_q$ is identically zero, so the second condition also doesn’t imply the first.