notes/docs/mathematics/functional-analysis/metric-spaces/completion.md

Completion

Definition 1: let (X,d) and (\tilde X, \tilde d) be metric spaces, then

1. a mapping T: X \to \tilde X is an isometry if \forall x, y \in X: \tilde d(Tx, Ty) = d(x,y).
2. (X,d) and (\tilde X, \tilde d) are isometric if there exists a bijective isometry T: X \to \tilde X.

Hence, isometric spaces may differ at most by the nature of their points but are indistinguishable from the viewpoint of the metric.

Or in other words, the metric space (\tilde X, \tilde d) is unique up to isometry.

Theorem 1: for every metric space (X,d) there exists a complete metric space (\tilde X, \tilde d) that contains a subset W that satisfies the following conditions

1. W is a metric space isometric with (X,d).
2. W is dense in X.

??? note "Proof:"

Will be added later.