Category:Completion Theorem (Normed Vector Space)

This category contains pages concerning 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.