Normed vector space

A normed vector space $(𝑉, 𝕂, ‖ \cdot ‖)$ is a Vector space over a Subfield $𝕂$ of $ℂ$ equipped with a norm $‖ \cdot ‖ : 𝑉 \to ℝ$ , where for any $𝑥, 𝑦 \in 𝑉$ and $𝛼 \in 𝕂$ , the following conditions hold: #m/def/anal/vec

Absolute homogeneity: $‖ 𝛼 𝑥 ‖ = | 𝛼 | ‖ 𝑥 ‖$
Triangle inequality: $‖ 𝑥 + 𝑦 ‖ \leq ‖ 𝑥 ‖ + ‖ 𝑦 ‖$
Positive-definite: $‖ 𝑥 ‖ = 0$ iff. $𝑥 = 0$ , otherwise $‖ 𝑥 ‖ > 0$

The norm of a vector generalises the idea of a vector's length. Every norm induces a metric over the vector space, where

𝑑 (⃗ 𝐯, ⃗ 𝐮) = ‖ ⃗ 𝐯 - ⃗ 𝐮 ‖

and consequently $‖ ⃗ 𝐯 ‖ = 𝑑 (⃗ 𝐯, ⃗ 𝟎)$ . The metric, in turn, induces a topology, making $𝑉$ a Topological vector space. A space which can be induced by a norm is called normable, but for example the norm for a normable metric is not unique in general. See Space for an overview of the relationship between different spaces.

By removing positive-definiteness, one gets a Seminormed vector space.

Properties

Equivalence of norms
A normed space is finite-dimensional iff the unit sphere is compact (use sequential compactness)

#state/tidy | #lang/en | #SemBr