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

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- Jun 30th 2009, 02:20 PMamro05Banach space proof
**Let E & F be a Banach spaces .Then E X F is Banach space??** - Jun 30th 2009, 02:30 PMHallsofIvy
That depends. If (u, v) is in $\displaystyle E\times F$, how do you define ||(u, v)||?

- Jun 30th 2009, 03:23 PMEnrique2
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.