TITLE: What is the coproduct in the category of Banach spaces and continuous linear maps?
QUESTION [7 upvotes]: In the category of Banach spaces, where the objects are Banach spaces and the morphisms are continuous linear maps, what are there coproducts? Are they the typical direct sum of Banach spaces? If so, does it accept finite and infinite coproducts?

REPLY [5 votes]: Yes, the coproduct of two Banach spaces $(X,\|~\|_X)$ and $(Y,\| ~ \|_Y)$ is $(X \oplus Y,\|~\|_{X\oplus Y})$ with $\|(x,y)\|_{X \oplus Y} = \|x\|_X + \|y\|_Y$. This is just an example for a suitable norm, equivalent norms such as $\sqrt{\|x\|_X^2 + \|y\|_Y^2}$ give isomorphic Banach spaces. The universal property is easily verified. It follows that finite coproducts exist. But infinite coproducts don't exist. I've proven this here recently for Hilbert spaces, cannot find the question right now (someone else?). The same argument applies here.