Consider C[0,1] with sup metric.

  • Thread starter Thread starter coverband
  • Start date Start date
  • Tags Tags
    Metric
Click For Summary
SUMMARY

The discussion focuses on the set of continuous functions, denoted as C[0, 1], and the application of the supremum metric to measure distances between these functions. The supremum metric quantifies the distance as the maximum absolute difference between two functions, f(x) and g(x), over the interval [0, 1]. It is established that since both functions are continuous, their difference, f - g, is also continuous, ensuring the existence of a maximum value on the closed interval. The conversation hints at exploring potential problems related to this metric.

PREREQUISITES
  • Understanding of continuous functions on closed intervals
  • Familiarity with metric spaces and distance metrics
  • Knowledge of supremum and maximum concepts in real analysis
  • Basic calculus, particularly dealing with functions and their properties
NEXT STEPS
  • Research the properties of metric spaces in real analysis
  • Explore examples of continuous functions in C[0, 1]
  • Study the implications of the supremum metric on convergence of functions
  • Investigate potential problems or paradoxes associated with supremum metrics
USEFUL FOR

Mathematicians, students of real analysis, and anyone interested in the properties of continuous functions and metric spaces.

coverband
Messages
170
Reaction score
1
jjjj
 
Physics news on Phys.org
Okay, I'm considering it!

(How long? I have a class to go to soon!)

For those who are wondering, "C[0, 1]" is the set of functions, f(x), that are continuous on the interval [0,1].

The "sup metric", also called "supremum metric" measures the "distance" between functions by the largest difference between values: max |f(x)- g(x)| over all x between 0 and 1. Notice that since f and g are continuous so is f- g and so the maximum over the closed and bounded interval [0, 1] does exist.

Of course, one would hope that there is some "problem" associated with this.
 

Similar threads

Undergrad Equivalence of alternative definitions of conservative vector fields and line integrals in different metric spaces
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Is the Sup Metric Used to Minimize Writing d_infinity in C[0,1]?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad How do non-diagonal indices of a metric allow for local flatness?
  • · Replies 22 ·
Replies
22
Views
2K
Undergrad Embedding Diagram of Weyl Metric in ##R^3##
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad Convergence not defined by any metric
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Raising and lowering indices using metric tensor
  • · Replies 9 ·
Replies
9
Views
874
Undergrad How to calculate basis vectors from metric tensor (as a matrix)?
  • · Replies 6 ·
Replies
6
Views
1K
Graduate Kruskal Coordinates in Schwartzchild metric
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad A sufficient condition for integrability of equation ##\nabla g=0##
  • · Replies 71 ·
3
Replies
71
Views
4K
Undergrad Metric Tensor on ##S^1## x ##S^2##
  • · Replies 6 ·
Replies
6
Views
2K