Thread: Banach space proof

Let E & F be a Banach spaces .Then E X F is Banach space??

That depends. If (u, v) is in , how do you define ||(u, v)||?

The answer is yes, the product topology of ExF can be defined with a norm, for instance sup(|| ||_E,|| ||_F), or suming the norms , or taking the root of the sum of the squares of the norms. All the "natural norms" (l_1, l_2 or l_\infty type) are equivalent.