Completion Theorem

Disambiguation

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Completion Theorem may refer to:

Completion Theorem (Metric Space)

Let $M = \struct {A, d}$ be a metric space.

Then there exists a completion $\tilde M = \struct {\tilde A, \tilde d}$ of $\struct {A, d}$.

Moreover, this completion is unique up to isometry.

That is, if $\struct {\hat A, \hat d}$ is another completion of $\struct {A, d}$, then there is a bijection $\tau: \tilde A \leftrightarrow \hat A$ such that:

$(1): \quad \tau$ restricts to the identity on $x$:

$\forall x \in A: \map \tau x = x$

$(2): \quad \tau$ preserves metrics:

$\forall x_1, x_2 \in A : \map {\hat d} {\map \tau {x_1}, \map \tau {x_2} } = \map {\tilde d} {x_1, x_2}$

Completion Theorem (Measure Space)

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Then there exists a completion $\struct {X, \Sigma^*, \bar \mu}$ of $\struct {X, \Sigma, \mu}$.

Completion Theorem (Normed Vector Space)

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Then there exists a Banach space $\struct {\widetilde X, \widetilde {\norm {\, \cdot \,} } }$ and a linear isometry $\phi : X \to \widetilde X$ such that $\phi \sqbrk X$ is dense in $\widetilde X$.

Further, the Banach space $\struct {\widetilde X, \widetilde {\norm {\, \cdot \,} } }$ is unique up to isometric isomorphism.