Conflicting definitions of Quadratic form in Hoffman Kunze linear algebra This question is about the exact definition of a quadratic form.
On page 273, the authors defined:

... quadratic form determined by the inner product is the function that assigns to each $\alpha$ the scalar $\|\alpha\|^2$...

On page 368, the authors defined:

Definition. Let $f$ be a bilinear form on the vector space V. We say that $f$ is symmetric $f(\alpha, \beta) = f(\beta, \alpha)$  for all $\alpha, \beta$ in $V$.


If $f$ is a symmetric bilinear form, the quadratic form associated with $f$ is the function $q$ from $V$ into $F$ defined by $q(\alpha)= f(\alpha, \alpha)$.

But complex Inner product is obviously not a bilinear form. Is there a typo or they are just two distinct concepts happening to use the same terminology?
Here is a related question.
 A: Neither of these sentences define a quadratic form. Here is a definition: if $V$ is a vector space over a field $K$, a quadratic form on $V$ is a function $q : V \to K$ such that $q(av) = a^2 q(v)$ for $a \in K$ and such that the function $q(u + v) - q(u) - q(v)$ is bilinear.
If $B : V \times V \to K$ is a symmetric bilinear form, you can check that $q(v) = B(v, v)$ is a quadratic form satisfying $q(u + v) - q(u) - q(v) = 2 B(u, v)$. If the characteristic of $K$ is not equal to $2$ then this gives a natural bijection between quadratic forms and symmetric bilinear forms.
Now let $V$ be a complex inner product space. The function $\| v \|^2 = \langle v, v \rangle$ is not a quadratic form over $\mathbb{C}$, because $ \| a v \|^2 = |a|^2 \| v \|^2$. This corresponds, as you say, to the complex inner product not being bilinear. However, it is a quadratic form over $\mathbb{R}$. In fact if $V = \mathbb{C}^n$ with the usual inner product it is a slightly disguised form of the usual Euclidean quadratic form on $\mathbb{R}^{2n}$. So the term "quadratic form" is still warranted here but one has to be a bit careful as you say.
