grandes-ecoles 2023 Q27

grandes-ecoles · France · x-ens-maths-a__mp Groups Ring and Field Structure

Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Conclude that $A$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ (Theorem C).

Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying
$$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$
Conclude that $A$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ (Theorem C).

Paper Questions

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