Let $n \geqslant 1$ be a natural number. We equip $\mathbb{R}^n$ with the usual inner product and the usual Euclidean norm defined, for all $X = (x_1, \ldots, x_n)$ and $Y = (y_1, \ldots, y_n)$ of $\mathbb{R}^n$, by $$(X \mid Y) = \sum_{i=1}^n x_i y_i \quad \text{and} \quad \|X\| = \sqrt{\sum_{k=1}^n x_k^2}$$ We study the existence of a bilinear map $B_n: (\mathbb{R}^n)^2 \rightarrow \mathbb{R}^n$ satisfying $$\forall X, Y \in \mathbb{R}^n, \quad \|B_n(X, Y)\| = \|X\| \times \|Y\|$$ Show the existence of such a bilinear map $B_n$ when $n$ is one of the integers $1, 2, 4$. For $n = 2$ (respectively 4) one may consider the product of two complex numbers (respectively of two quaternions).
Let $n \geqslant 1$ be a natural number. We equip $\mathbb{R}^n$ with the usual inner product and the usual Euclidean norm defined, for all $X = (x_1, \ldots, x_n)$ and $Y = (y_1, \ldots, y_n)$ of $\mathbb{R}^n$, by
$$(X \mid Y) = \sum_{i=1}^n x_i y_i \quad \text{and} \quad \|X\| = \sqrt{\sum_{k=1}^n x_k^2}$$
We study the existence of a bilinear map $B_n: (\mathbb{R}^n)^2 \rightarrow \mathbb{R}^n$ satisfying
$$\forall X, Y \in \mathbb{R}^n, \quad \|B_n(X, Y)\| = \|X\| \times \|Y\|$$
Show the existence of such a bilinear map $B_n$ when $n$ is one of the integers $1, 2, 4$.
For $n = 2$ (respectively 4) one may consider the product of two complex numbers (respectively of two quaternions).