Let $X$ and $Y$ be two normed vector spaces and $n(\cdot, \cdot)$ be any norm on $\mathbb{R}^2$. Is it always possible to define a norm on the product vector space $X \times Y$ as $||(x, y)||\_{X \times Y} = n(||x||\_X, ||y||\_Y)$?


*Background information:* the book "Advanced Calculus" by Sternberg and Loomis says the answer is negative - you have to impose an additional requirement on $n(\cdot, \cdot)$ for that to hold. However, no example is given in the text nor the exercises. Can you give me an example of such a pathological norm $n$ that fails to induce a honest normed vector space on $X \times Y$.



Pathological product space norm
Let $X=Y=\mathbb R$ with the absolute value norm and define $n(a,b)=\sqrt{2a^2+2b^2-3ab}$. This is a norm on $\mathbb R^2$ because it is the quadratic form of the positive definite matrix $A=\left( \begin{smallmatrix} 2 & -3/2 \\ -3/2 & 2 \end{smallmatrix}\right)$.


Then $N(v) = n(|v\_1|,|v\_2|)$ is not a norm because the triangle inequality fails for $u=(1,1)$, $v=(1,-1)$: we have that $N(u)=N(v)=1$, but $N(u+v)=2\sqrt{2}$.


