TITLE: Pathological product space norm
QUESTION [10 upvotes]: 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 $X \times Y$.

REPLY [7 votes]: This is a comment rather than an answer but I am not entitled.  The missing requirement on the two dimensional norm for the statement to be valid is that it be increasing in the natural sense that if $|x|\leq|x_1|, |y|\leq|y_1|$, then $n(x,y)\leq n(x_1,y_1)$.  Norms without this property can easily be obtained by rotating a non-circular ellipse in standard position (as in the above answer).