My question : Suppose X is an infinite dimensional normed linear space. Does there exist any normed topology on $X \times X$ such that the projection maps (i.e. $(x,y) \mapsto x$ and $(x,y) \mapsto y$) are not continuous.
My idea : First of all, for any $p \in [1,\infty]$ if we define $\|. \|_p$ on $X \times X$ as $\|(x, y)\|_p = (\|x\|^p + \|y\|^p) ^{\frac{1}{p}}.$ Then it will not satisfy above criteria.
To construct desired norm we should impose the following condition on that norm :
there exists $(x, y)\in X \times X$ such that either $\| x\|>\|(x, y)\|$ or $\| y\|>\|(x, y)\|.$
I don't know if it is possible at all.
But since X is infinite dimensional so $X \times X$ is also infinite dimensional and it is known that on every infinite dimensional normed space there exist non-equivalent norms. So, on $X \times X$ there exists a norm such that induced topology is not equivalent to product topology. But can there be any norm on $X \times X$ so that induced topology is weaker than the product topology?
Best Answer
Denote with $(X\times X)_2$ a Banach structure on $X\times X$. Suppose that both $\pi_1,\pi_2$ are continuous on $(X\times X)_2$, then the map:
$$(X\times X)_2\to X\times X, \qquad x\mapsto (\pi_1(x), \pi_2(x))$$ is a continuous bijection between Banach spaces and so by the open mapping theorem its an isomorphism. But on a given infinite dimensional Banach spaces there exist many Banach norms on it that cannot be isomorphic to the original one. Take any one of those and you get a Banach space structure on the product for which at most one of the projections can be continuous.
If you want both projections to be discontinuous use $X_2$ to denote the vectorspace $X$ equipped with a Banach norm that is inequivalent to the given norm on $X$. Then $X_2\times X_2$ is the same vectorspace as $X\times X$, but now if one of the projections is continuous (say eg that $\pi_1$ is continuous) then the restriction of $\pi_1$ to $X_2\times\{0\}$ remains continuous. But now we have built a bijective continuous map between the Banach space $X_2$ and $X$ - impossible. So in this case both projections are discontinuous.