Definition:Subdivision (Real Analysis)/Finite



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  • 2 Also known as
  • 3 Sources

    • Definition

    Let $\closedint a b$ be a closed interval of the set $\R$ of real numbers.


    Let $x_0, x_1, x_2, \ldots, x_{n - 1}, x_n$ be points of $\R$ such that:

    $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$


    Then $\set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a finite subdivision of $\closedint a b$.


    Normal Subdivision

    $P$ is a normal subdivision of $\closedint a b$ if and only if:

    the length of every interval of the form $\closedint {x_i} {x_{i + 1} }$ is the same as every other.


    That is, if and only if:

    $\exists c \in \R_{> 0}: \forall i \in \N_{< n}: x_{i + 1} - x_i = c$


    Also known as

    Some sources use the term partition for the concept of a subdivision.

    However, the latter term has a different and more general definition, so its use is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.

    Some use the term dissection, but again this also has a different meaning, and is similarly discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.


    Some sources do not define the concept of infinite subdivision, and so simply refer to a finite subdivision as just a subdivision.


    Sources

    Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Subdivision_(Real_Analysis)/Finite&oldid=665812"

    Category: 
    ●Definitions/Subdivisions (Real Analysis)


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