Category:Completion Theorem
This category contains pages concerning Completion Theorem:
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.
Subcategories
This category has the following 3 subcategories, out of 3 total.
Pages in category "Completion Theorem"
The following 4 pages are in this category, out of 4 total.