Topology/Vector Spaces

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

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

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

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

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

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

5. Identity element (addition): $$\exists$$ 0 $$\in V$$ such that u+0 = u $$\forall$$ u $$\in V$$

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

7. Commutativity: u+v = v+u

8. Inverse element: $$\exists$$ -u $$\in V$$ such that u+(-u) = 0 $$\forall$$ u $$\in V$$