grandes-ecoles 2023 Q26

grandes-ecoles · France · x-ens-maths-a__mp Groups Algebra and Subalgebra Proofs

Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Show that $x^2 \in \mathbb{R} + \mathbb{R}x$ for all $x \in A$. One may use the result from question 25 with $y = 1$.

Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying
$$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$
Show that $x^2 \in \mathbb{R} + \mathbb{R}x$ for all $x \in A$. One may use the result from question 25 with $y = 1$.

Paper Questions

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