Normed vector space

Builds on Norm

Definition

Let $\mathbb{K}$ be $\R$ or $\C$ . A normed vector space over $\mathbb{K}$ is a vector space $V$ over $\mathbb{K}$ together with a norm $\norm{\cdot}$ on $V$ .

Remarks

A normed space is a metric space with the distance $d(x, y) = \norm{x - y}$ . A normed space that is complete for this metric is a Banach space. Each inner product space is a normed space with the induced norm. On a finite-dimensional space, all norms are equivalent, so they define the same topology.

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Vector space definition
Norm definition
Normed vector space definition

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6 pages build on this one.

Tag	`00A0RPAQ8A`
URI	`m2://00A0RPAQ8A`
Kind	definition
Writing	W1 stated
MSC	46B20
Defines	Normed vector space
Uses	Norm

Normed vector space

Definition

Remarks

Reading path

Where this leads

Further reading

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