📂Banach Space

Isometric Mapping

Definitions

Given two metric spaces $(X,\ d_X), (Y,\ d_Y)$, if there exists a mapping $f : X \to Y$ that satisfies the conditions below, then $X$ and $Y$ are said to be isometric, denoted by $X \approx Y$. Furthermore, the mapping $f$ is called an isometric map or isometry.

$$d_X(x_1,\ x_2) =d_Y\big( f(x_1),\ f(x_2) \big),\quad \forall\ x_1,x_2\in X$$

Explanation

As the name suggests, an isometric map is a mapping that preserves distance. Therefore, two spaces that have an isometric map between them can be considered ’essentially’ the same. Moreover, an isometry naturally becomes a one-to-one function from its definition.

In Normed Spaces

If $X$ and $Y$ are normed spaces, since the distance is defined below, an isometric map becomes a mapping that preserves the norm.

$$d_X(x_1,x_2) = \|x_1-x_2\|_X$$

Definition¹

Let $(X, \left\| \cdot \right\|_{X}), (Y, \left\| \cdot \right\|_{Y})$ be a normed space. If there exists a linear operator $L\ : X \to Y$ that satisfies the conditions below for $X$ and $Y$, then $L$ is called an isometric isomorphism. Moreover, $X$ and $Y$ are said to be isometrically isomorphic.

$$\|x\|_X = \|L(x)\|_Y, \quad \forall\ x\in X$$

Properties

The following facts hold for isometric maps:

• Isometric maps are one-to-one functions.
• Isometric maps are homeomorphisms.
• $\approx$ is an equivalence relation.
• An isometric map is an embedding.

Proof

Let $x_1,x_2\in X$ and $f(x_1)=f(x_2)$. Then, by the definition of distance, $d_Y\big( f(x_1),\ f(x_2) \big)=0$ holds. Since $f$ preserves distance, $d_X(x_1,\ x_2)=0$ holds and similarly by the definition of distance, $x_1=x_2$ holds. If $f(x_1)=f(x_2)$, then $x_1=x_2$ holds, hence $f$ is a one-to-one function.

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