Let $(V, ||\cdot||)$ be a normed vector space (not necessarily finite-dimensional) and consider the product topology on $V \times V$.
Why is the product topology on $V \times V$ the topology generated by the usual Euclidean metric?
The topology induced on $V$ by the norm $||\cdot||$, certainly doesn't have to be the Euclidean one, and my definition of product topology is the smallest topology for which the projection maps are continous.
Can anybody help me understand why the above is true? Thank you.