X x Y is a Banach Space: Conditions Explored

Click For Summary
SUMMARY

X x Y is a Banach space if and only if both X and Y are Banach spaces. The norm on X x Y is defined as ||(x,y)|| = ||x|| + ||y||, where x belongs to X and y belongs to Y. A normed space is classified as a Banach space if it is complete, meaning every Cauchy sequence in the space converges to a limit within the space. Understanding the implications of completeness and the relationship between sequences in X and X x Y is crucial for grasping the conditions under which X x Y maintains its Banach space properties.

PREREQUISITES
  • Understanding of normed linear spaces (nls)
  • Familiarity with the concept of completeness in metric spaces
  • Knowledge of Cauchy sequences and their convergence
  • Basic definitions and properties of Banach spaces
NEXT STEPS
  • Study the properties of Cauchy sequences in normed spaces
  • Explore the concept of completeness in metric spaces
  • Investigate examples of Banach spaces and their norms
  • Learn about the implications of product spaces in functional analysis
USEFUL FOR

Mathematicians, students of functional analysis, and anyone studying properties of Banach spaces and their applications in various mathematical contexts.

Ant farm
Messages
19
Reaction score
0
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
 
Physics news on Phys.org
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.
 

Similar threads

Graduate What separates Hilbert space from other spaces?
  • · Replies 59 ·
2
Replies
59
Views
7K
Graduate Hilbert, Banach and Fourier theory
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Is the Quotient of a Banach Space by a Closed Linear Subspace also Banach?
  • · Replies 1 ·
Replies
1
Views
3K
MHB Understanding Example from Topics in Banach Space Integration
  • · Replies 2 ·
Replies
2
Views
2K
Show that a space is a Banach space
  • · Replies 12 ·
Replies
12
Views
2K
MHB Proving that $X$ is a Banach Space and $Y$ is Not
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Can Quantum Mechanics be Studied in Banach Spaces or Other Non-Hilbert Spaces?
  • · Replies 17 ·
Replies
17
Views
2K
Graduate Lagrange multipliers on Banach spaces (in Dirac notation)
  • · Replies 2 ·
Replies
2
Views
2K
High School Parametric Representation of Lines
  • · Replies 13 ·
Replies
13
Views
3K
Graduate A Holder space is a Banach space
  • · Replies 12 ·
Replies
12
Views
5K