Doubt in proof of Jordan - von Neumann theorem

Question

In Paliogiannis and Martin A. Moskowitz's Functions Of Several Real Variables, the Jordan - von Neumann theorem is given as follows: The inner product is defined as: $$\langle x,y \rangle = \frac{||x+y||^2-||x-y||^2}{4}$$ For proving $\langle cx,y \rangle = c\langle x,y \rangle $, the book describes the following step: The reasoning behind it is not clear to me. I am aware of the method which was used in original paper to prove this ie. first proving it for $c \in \mathbb{Q}$ and using the continuity of norm and density of $\mathbb{Q}$ to prove for $c \in \mathbb{R}$. But I wanted to see if there is an easier way which the book intends to convey. I tried a lot of different manipulations but to no avail.
I'm just curious whether there is some other method which this book wants to convey or is it just a genuine oversight? I found an errata for this book, but it had no mention of this theorem.

Answer 1

The way he proved $\langle cx, y \rangle = c \langle x, y \rangle$, for $c \in \mathbb{R}$

In this step: $\langle cx, y\rangle = \frac14 [||cx+y||^2 - ||cx- y ||^2]$. The author used his definition of the inner product.

As for $\frac14 [||cx+y||^2 - ||cx- y ||^2] = \frac14 [4c\langle x, y\rangle]$, It seems there is a gap there. We need to evoke the parallelogram law to reach there.