X x Y is a Banach Space: Conditions Explored

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Discussion Overview

The discussion revolves around the conditions under which the Cartesian product of two normed spaces, X and Y, forms a Banach space. Participants explore the definitions and implications of completeness in normed spaces and the specific norms applied to the product space.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant suggests that X x Y is a Banach space if and only if both X and Y are Banach spaces.
  • Another participant inquires about the specific norm on X x Y in relation to the norms of X and Y, and seeks clarification on the definition of a Banach space.
  • A participant specifies that X and Y are normed linear spaces (nls) with the regular norm and defines the norm on X x Y as ||(x,y)|| = ||x|| + ||y||.
  • There is a question regarding the meaning of completeness and how to demonstrate that sequences in X can be related to sequences in X x Y, as well as the reverse.
  • One participant comments on the nature of previous questions posed, indicating they were not merely for clarification but essential for addressing the main question.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are multiple inquiries and points of clarification regarding the definitions and implications of completeness and norms in the context of Banach spaces.

Contextual Notes

There are unresolved aspects regarding the completeness of the product space and the specific conditions under which the properties of X and Y influence the structure of X x Y.

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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
 
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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?
 
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.
 
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?
 
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.
 

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