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 Shlomo 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 $X \times Y$.

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 $X \times Y$.