Partition of an infinite-dimensional interval

  • Context: Graduate 
  • Thread starter Thread starter tpm
  • Start date Start date
  • Tags Tags
    Interval Partition
Click For Summary
SUMMARY

This discussion focuses on the partitioning of infinite-dimensional intervals, specifically within functional spaces. The proposed method involves a bisection-like approach to create partitions centered around a specific function, denoted as X(t). The formula presented, X(t) + nδ(t - t'), illustrates how to generate these partitions, where δ represents a perturbation factor. The conclusion emphasizes that the dimensionality of the space does not affect the ability to create infinite-dimensional partitions.

PREREQUISITES
  • Understanding of functional spaces and infinite-dimensional intervals
  • Familiarity with partition theory in mathematics
  • Knowledge of perturbation methods in functional analysis
  • Basic grasp of mean value concepts in mathematical analysis
NEXT STEPS
  • Research "functional analysis and infinite-dimensional spaces"
  • Explore "partition theory in mathematical contexts"
  • Study "perturbation methods in functional analysis"
  • Learn about "mean value theorems in analysis"
USEFUL FOR

Mathematicians, researchers in functional analysis, and students studying advanced mathematical concepts related to infinite-dimensional spaces and partition theory.

tpm
Messages
67
Reaction score
0
Is ther any form by bisection or simiar to obtain a partition for an infinite-dimensional interval (aka functional space)?? i believe you could obtain a partition for every interval 'centered' at a certain function as:

[tex]X(t), X(t)+\delta (t-t`), X(t)+2\delta (t-t`), X(t)+\delta (t-t`), ...[/tex]

where X(t) is the center of the partition in the sense that the mean value of..

[tex]\sum_{i=1}^{N} \frac{ Y_{i} (t)}{N}=X(t)[/tex]

where the Y_i (t) are the 'elements' of the partition...
 
Physics news on Phys.org
The dimension is irrelevant for partitions. You just get infinite dimensional parts.
 

Similar threads

Undergrad Express the limit as a definite integral
  • · Replies 16 ·
Replies
16
Views
4K
Undergrad Attempting to find an intuitive proof of the substitution formula
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Finding the derivative of a characteristic function
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Lipschitz continuity of vector-valued function
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Summing over continuum and uncountable numerocities
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Definitions of the Riemann integral
  • · Replies 14 ·
Replies
14
Views
3K
Graduate The partial derivative of a function that includes step functions
  • · Replies 3 ·
Replies
3
Views
4K
Partition function from the density of states
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Compact summary of Fourier series equations
  • · Replies 11 ·
Replies
11
Views
2K
Graduate Inverse function of the Nyquist-Shannon sampling theorem
  • · Replies 2 ·
Replies
2
Views
2K