When is X × Y a Banach space?

[Ant farm]

Hey there,

could you guide me in the following question:

X x Y is a Banach space if and only if X and Y are both Banach Spaces

Thank you

 

[matt grime]

What is the norm on XxY in terms of the norms of X and Y? What does it mean for a normed space to be a banach space?

 

[Ant farm]

Sorry, wasn't clear,

X, Y nls's with the regular norm.
X x Y an nls with norm ||(x,y)|| = ||x|| + ||y||, x belonging to X and y belonging to Y.

A normed space is a banach space if it is a complete nls.

 

[matt grime]

So what does complete mean? Can you think of a way to show that given a sequence in X, you can get one in XxY? And given a sequence in XxY how does one get a sequence in X? Now what about using the hypotheses?

 

[HallsofIvy]

matt grimes' questions, both in his first response and in his second, were not asking for clarification. Those are the questions you need to think about in order to answer your question.