Analysis MOC

Normed vector space

A normed vector space (𝑉,𝕂,‖ ⋅‖) is a Vector space over a Subfield 𝕂 of ℂ equipped with a norm ‖ ⋅‖ :𝑉 →ℝ, where for any 𝑥,𝑦 ∈𝑉 and 𝛼 ∈𝕂, the following conditions hold: vec

1. Absolute homogeneity: ‖𝛼𝑥‖ =|𝛼|‖𝑥‖
2. Triangle inequality: ‖𝑥 +𝑦‖ ≤‖𝑥‖ +‖𝑦‖
3. 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)

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