# Topology/Vector Spaces

A **vector space** is formed by scalar multiples of vectors. The scalars are most commonly real numbers but can also be complex, or from any field.

## Definition[edit]

A vector space on a field is a set equipped with two binary operations: common vector addition on elements of and scalar multiplication, by elements of on elements of . These operations are subject to 8 axioms (**u**,**v**, and **w** are vectors in and a and b are scalars in ):

1. Associativity (addition): (**u**+**v**)+**w** = **u**+(**v**+**w**).

2. Associativity (scalar and field multiplication): a(b**u**) = (ab)**u**

3. Distributivity (field addition): (a+b)**u** = a**u**+b**u**

4. Distributivity (vector addition): a(**u**+**v**) = a**u**+a**v**

5. Identity element (addition): **0** such that **u**+**0** = **u** **u**

6. Identity element (scalar multiplication): 1**u** = **u**

7. Commutativity: **u**+**v** = **v**+**u**

8. Inverse element: -**u** such that **u**+(-**u**) = **0** **u**