Multiplicative norm on $\mathbb{R}[X]$.

How to prove that : there is no function $N\colon \mathbb{R}[X] \rightarrow \mathbb{R}$, such that : $N$ is a norm of $\mathbb{R}$-vector space and $N(PQ)=N(P)N(Q)$ for all $P,Q \in \mathbb{R}[X]$.

Once, my teacher asked if there is a multipicative norm on $\mathbb{R}[X]$, and one of my classmate proved that there was none. But I can't remember the proof (all I remember is that he was using integration somewhere...).

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Remark: Such $N$ is completely determined by the values $N(X+a)$ with $a\in\mathbb R$ and $N(X^2+bX+c)$ with $b^2<4c$. – Hagen von Eitzen Oct 28 '12 at 14:43
Here is a (perhaps stupid) idea: Assume by contradiction that such a norm exists. Show that the completion of $\mathbb R[X]$ is a Banach algebra isomorphic to the Banach algebra of continuous functions on a compact Hausdorff space, and derive a contradiction from that. – Pierre-Yves Gaillard Nov 3 '12 at 12:25
@Pierre-YvesGaillard: I think the idea with completion is a good one. Instead of trying to prove that $\mathbb{R}[X]$ is an algebra of continuous functions (which is non-trivial, see e.g. this American Math Monthly paper) it would probably be easier to appeal to Mazur's theorem stating that the only (associative, unital) real Banach division algebras are $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$. Davide pointed out that my answer also answers the present question. It seems far too complicated... – commenter Nov 3 '12 at 14:41
@Pierre-YvesGaillard: I'm not sure if your argument uses the fact that the norm is multiplicative. It seems that it only requires the norm to be sub-multiplicative. – Haskell Curry Nov 4 '12 at 11:33
@commenter I agree the approach to the problem you linked works here and seems too complicated. At least, it shows that such a norm doesn't exist, so we are looking to a quite simple argument showing that. – Davide Giraudo Nov 4 '12 at 12:03

Using the article Absolute-Valued Algebras mentioned by commenter, a proof would look like this.

Firstly, recall

Lemma If $(E,\| \cdot \|)$ is a normed $\mathbb{R}$-vector space such that $\| x + y \|^{2} + \| x - y \|^{2} \geq 4 \| x \| \| y \|$ holds for all $x,y \in E$, then the norm $\| \cdot \|$ is induced by some inner product $\langle \cdot,\cdot \rangle$ on $E$.

Note Anyone who has a link to a proof of the lemma above is warmly welcome.

Suppose that $\| \cdot \|$ is a multiplicative norm on $\mathbb{R}[X]$. Then for any $x,y \in \mathbb{R}[X]$, we have \begin{align} \| x + y \|^{2} + \| x - y \|^{2} &= \| (x + y)^{2} \| + \| (x - y)^{2} \| \quad (\text{By the multiplicativity of $\| \cdot \|$.}) \\ &\geq \| (x + y)^{2} - (x - y)^{2} \| \quad (\text{By the Triangle Inequality.}) \\ &= \| 4xy \| \\ &= 4 \| x \| \| y \|. \quad (\text{By the multiplicativity of $\| \cdot \|$.}) \end{align} The lemma now says that $\| \cdot \|$ is induced by some inner product $\langle \cdot,\cdot \rangle$ on $\mathbb{R}[X]$.

By applying the Gram-Schmidt orthogonalization procedure to the linearly independent set $\{ 1,X \}$, we obtain a non-zero $a \in \mathbb{R}[X]$ that is orthogonal to $1$ (namely, $a = X - \frac{\langle 1,X \rangle}{\langle 1,1 \rangle} \cdot 1$). Then \begin{align} \| 1 - a^{2} \| &= \| (1 - a)(1 + a) \| \\ &= \| 1 - a \| \| 1 + a \| \quad (\text{By the multiplicativity of $\| \cdot \|$.}) \\ &= \| 1 - a \|^{2} \quad (\text{By orthogonality, $\| 1 + a \| = \| 1 - a \|$.}) \\ &= \langle 1 - a,1 - a \rangle \\ &= \| 1 \|^{2} + \| a \|^{2} - 2 \langle 1,a \rangle \\ &= \| 1 \|^{2} + \| a \|^{2} \quad (\text{By orthogonality once again.}) \\ &= \| 1 \| + \| a^{2} \| \quad (\text{By the multiplicativity of $\| \cdot \|$.}) \\ &= \| 1 \| + \| - a^{2} \|. \end{align} In summary, $\| 1 - a^{2} \| = \| 1 \| + \| - a^{2} \|$ holds. Squaring both sides of this equation gives us \begin{align} \| 1 - a^{2} \|^{2} &= (\| 1 \| + \| - a^{2} \|)^{2}, \\ \langle 1 - a^{2},1 - a^{2} \rangle &= \| 1 \|^{2} + \| - a^{2} \|^{2} + 2 \| 1 \| \| - a^{2} \|, \\ \| 1 \|^{2} + \| - a^{2} \|^{2} + 2 \langle 1,- a^{2} \rangle &= \| 1 \|^{2} + \| - a^{2} \|^{2} + 2 \| 1 \| \| - a^{2} \|, \\ \langle 1,- a^{2} \rangle &= \| 1 \| \| - a^{2} \|. \end{align} Hence, we get equality when we apply the Cauchy-Schwarz Inequality to the vectors $1$ and $- a^{2}$. Equality holds if and only if both $1$ and $- a^{2}$ lie in the same $1$-dimensional subspace of $\mathbb{R}[X]$; as $1,- a^{2} \neq 0$, this says that both are non-zero scalar multiples of each other. However, $\langle 1,- a^{2} \rangle = \| 1 \| \| - a^{2} \| > 0$, so more precisely, $1$ and $- a^{2}$ are positive scalar multiples of each other. We therefore obtain $a^{2} \in \mathbb{R}_{<0}$, which is a contradiction. ////

Conclusion There can be no multiplicative norm on $\mathbb{R}[X]$.

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A variant of the lemma (assuming $|x| = |y| = 1$) is a theorem due to Schoenberg, A remark on M. M. Day's characterization of inner-product spaces and a conjecture of L. M. Blumenthal, Proc. Amer. Math. Soc. 3 (1952), 961-964, Theorem 2. There are some squares missing in the last displayed equation, I think you want $|1-a^2|^{2} = \cdots = |1|^2 + |\pm a^2|^2$. – commenter Nov 8 '12 at 17:52
@commenter That last equation seems just fine as it is? – WimC Nov 8 '12 at 21:21
@WimC: Oh, yes, you're right. Too much switching back and forth between squares and square roots :-) Sorry for the false alarm. – commenter Nov 8 '12 at 21:54
Hi folks. The recent edit was made in order to address the issue of clarity mentioned in the comments here. – Haskell Curry Dec 21 '12 at 6:26