Editing Euclidean space (section)

===Subspaces===
{{main|Flat (geometry)}}
Let {{mvar|E}} be a Euclidean space and <math>\overrightarrow E</math> its associated vector space.

A ''flat'', ''Euclidean subspace'' or ''affine subspace'' of {{mvar|E}} is a subset {{mvar|F}} of {{mvar|E}} such that

<math display="block">\overrightarrow F = \Bigl\{\overrightarrow {PQ} \mathrel{\Big|} P\in F, Q\in F \Bigr\}\vphantom{\frac({}}</math>

as the associated vector space of {{mvar|F}} is a [[linear subspace]] (vector subspace) of <math>\overrightarrow E.</math> A Euclidean subspace {{mvar|F}} is a Euclidean space with <math>\overrightarrow F</math> as the associated vector space. This linear subspace <math>\overrightarrow F</math> is also called the ''direction'' of {{mvar|F}}.

If {{mvar|P}} is a point of {{mvar|F}} then

<math display="block">F = \Bigl\{P+v \mathrel{\Big|} v\in \overrightarrow F \Bigr\}.</math>

Conversely, if {{mvar|P}} is a point of {{mvar|E}} and <math>\overrightarrow V</math> is a [[linear subspace]] of <math>\overrightarrow E,</math> then

<math display="block">P + \overrightarrow V = \Bigl\{P + v \mathrel{\Big|} v\in \overrightarrow V \Bigr\}</math>

is a Euclidean subspace of direction <math>\overrightarrow V</math>. (The associated vector space of this subspace is <math>\overrightarrow V</math>.)

A Euclidean vector space <math>\overrightarrow E</math> (that is, a Euclidean space that is equal to <math>\overrightarrow E</math>) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.