Definition:Isometric Isomorphism/Normed Vector Space
< Definition:Isometric Isomorphism(Redirected from Definition:Isometrically Isomorphic Normed Vector Spaces)
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Definition
Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.
Let $T : X \to Y$ be a linear isometry.
We say that $T$ is an isometric isomorphism if and only if $T$ is bijective.
If an isometric isomorphism $T : X \to Y$ exists, we say that $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ are isometrically isomorphic.
Also see
- Results about isometric isomorphisms on normed vector spaces can be found here.
Sources
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $3.5$: Isomorphisms between Normed Spaces