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Mathematics for Physics

343 Completeness

A set where every sequence of elements that get closer to each other has a limit in the set that is used for finding the limit of a function.

definition [d] (Completeness) From Kreyszig Definition 1.4-3: the space \(X\) is said to be complete if every Cauchy sequence in \(X\) converges, that is, has a limit which is an element of \(X\).

where

  • \(X = (X,d)\) is a metric space.
  • a Cauchy sequence is as in Kreyszig Definition 1.4-3.

Note:

  • Kreyszig Theorem 1.4-4: the real line and the complex plane are complete metric spaces.

343.1 Examples

343.1.1 Simple

A metric space is complete when every Cauchy sequence converges in the space. The real line is complete.

\[ X = \mathbb{R},\quad d(x,y) = |x - y| \]

\[ x_{n} = \dfrac{1}{n} \text{ is Cauchy and } x_{n} \to 0 \in \mathbb{R} \]

where

  • \(X\) is the metric space.
  • completeness means every Cauchy sequence has a limit in \(X\).

343.1.2 General

The plane \(\mathbb{R}^{2}\) with the Euclidean metric is complete: every Cauchy sequence of points converges to a point of \(\mathbb{R}^{2}\).

\[ X = \mathbb{R}^{2},\quad d(x,y) = \sqrt{(x_{1}-y_{1})^{2} + (x_{2}-y_{2})^{2}} \]

where

  • Kreyszig records that \(\mathbb{R}\) and \(\mathbb{C}\) are complete; the same holds for \(\mathbb{R}^{n}\).