Analysis MOC

Normed vector space

A normed vector space is a Vector space over a Subfield of equipped with a norm , where for any and , the following conditions hold: vec

1. Absolute homogeneity:
2. Triangle inequality:
3. Positive-definite: iff. , otherwise

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

and consequently . The metric, in turn, induces a topology, making 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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