= Dimension bound for a commutative Euclidean normed algebra
{title2=$\dim_{\mathbb R}A\leq2$}
A finite-dimensional <commutative> real algebra with a Euclidean multiplicative <norm> has dimension at most two. <Automatic real bilinearity from a multiplicative norm> supplies bilinearity if it was not part of the initial description. On its <unit sphere>, squaring has derivative $v\mapsto2xv$, injective because there are no <zero divisors>. The only equal squares are antipodal pairs, so the map induces an injective <immersion> $\mathbb{RP}^{n-1}\to S^{n-1}$. <Compactness> and the <inverse function theorem> make it a <diffeomorphism> when $n\geq2$. For $n\geq3$, the <fundamental groups> $\mathbb Z/2$ and zero contradict this. The real and complex number algebras attain dimensions one and two.
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