Definition:Subdivision (Real Analysis)/Finite
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
- 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.3$ Definitions
- 1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): $\S 6.3$: Functions of Bounded Variation: Definition $6.3$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.2$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): partition: 2. (of an interval)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): partition: 2. (of an interval)
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $2.5$: The Riemann Integral
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Lengths and Distances