Noun

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.

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^*:V→V, 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 $\backslash mathbb\{F\}$ is a one-dimensional vector space over itself.
   • If $\backslash mathbb\{V\}$ is a vector space over $\backslash mathbb\{F\}$ and S is any set, then $\backslash mathbb\{V\}^S=\backslash \{f|f:S\backslash rightarrow\; \backslash mathbb\{V\}\; \backslash \}$ is a vector space over $\backslash mathbb\{F\}$, and $\backslash mbox\{dim\}\; (\; \backslash mathbb\{V\}^S\; )\; =\; \backslash mbox\{card\}(S)\; \backslash ,\; \backslash mbox\{dim\}\; (\backslash mathbb\{V\})$.
   • If $\backslash mathbb\{V\}$ is a vector space over $\backslash mathbb\{F\}$ then any closed subset of $\backslash mathbb\{V\}$ is also a vector space over $\backslash mathbb\{F\}$.
   • The above three rules suffice to construct all vector spaces.