# Banach space proof

• Jun 30th 2009, 03:20 PM
amro05
Banach space proof
Let E & F be a Banach spaces .Then E X F is Banach space??
• Jun 30th 2009, 03:30 PM
HallsofIvy
That depends. If (u, v) is in $E\times F$, how do you define ||(u, v)||?
• Jun 30th 2009, 04:23 PM
Enrique2
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.