Norm-Induced Metric Space

analysis topology

Definition

Norm-Induced Metric Space

A norm-induced metric space is the metric space obtained from a normed vector space by using the norm to define distance.

Construction

Let $⟨V,\Vert \cdot \Vert ⟩$ be a normed vector space. The induced metric space is $⟨V,d⟩$, where

$$d(x,y)=\Vert x-y\Vert .$$

The underlying set is still $V$. The norm supplies the distance between two vectors by measuring the length of their difference.