Vector space

# Vector space

Wiktonary

### Noun

vector space
1. A type of set of vectors that satisfies a specific group of constraints.
A vector space is a set of vectors which can be linearly combined.
Each vector space has a basis and dimension.

vector space over the field F
1. A set V, whose elements are called "vectors", together with a binary operation + forming a module (V,+), and a set F* of bilinear unary functions f*:VV, each of which corresponds to a "scalar" element f of a field F, such that the composition of elements of F* corresponds isomorphically to multiplication of elements of F, and such that for any vector v, 1*(v) = v.
• Any field $\mathbb\left\{F\right\}$ is a one-dimensional vector space over itself.
• If $\mathbb\left\{V\right\}$ is a vector space over $\mathbb\left\{F\right\}$ and S is any set, then $\mathbb\left\{V\right\}^S=\\left\{f|f:S\rightarrow \mathbb\left\{V\right\} \\right\}$ is a vector space over $\mathbb\left\{F\right\}$, and $\mbox\left\{dim\right\} \left( \mathbb\left\{V\right\}^S \right) = \mbox\left\{card\right\}\left(S\right) \, \mbox\left\{dim\right\} \left(\mathbb\left\{V\right\}\right)$.
• If $\mathbb\left\{V\right\}$ is a vector space over $\mathbb\left\{F\right\}$ then any closed subset of $\mathbb\left\{V\right\}$ is also a vector space over $\mathbb\left\{F\right\}$.
• The above three rules suffice to construct all vector spaces.