Definition:Normed Vector Space

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Let $\struct {K, +, \circ}$ be a normed division ring.

Let $V$ be a vector space over $K$.

Let $\norm {\,\cdot\,}$ be a norm on $V$.

Then $\struct {V, \norm {\,\cdot\,} }$ is a normed vector space.

Also known as

When $\norm {\,\cdot\,}$ is arbitrary or not directly relevant, it is usual to denote a normed vector space merely by the symbol $V$.

A normed vector space is also known as a normed linear space.

Some sources refer just to a normed space.

Also see

  • Results about normed vector spaces can be found here.