Vector Space

In mathematics, vector space is a set of objects that can be operated(added, multipled and so on) together with number

A vector space is a triple(V, +, .) and it must obey the following 9 rules:

1. Closure by addition: \[ \forall u, w \in V, \quad u+w \in V \]
2. Additive associality: \[ \forall u, v, w \in V, \quad (u+v)+w=u+(v+w) \]
3. Additive commutativity: \[ \forall u, w \in V, \quad u+w = w+u \]
4. Additive identity: \[ \exists 0 \in V, \quad u+0 = 0+u \]
5. Additive inverse: \[ \forall u \in V, \quad -u \in V \quad s.t. \quad (-u)+u=u+(-u)=0 \]
6. Closure by scalar multiplication: \[ \forall k \in R \quad and \quad u \in V, \quad k*u \in V \]
7. Multiplicative associality: \[ \forall k,l \in R \quad and \quad u \in V, \quad (k*l)*u=k*(l*u) \]
8. Multiplicative identity: \[ \exists 1 \in V, \quad 1*u=u\]
9. Distributive property: \[ \forall k \in R \quad and \quad u,w \in V, \quad k*(u+w)=k*u+k*w \] \[ \forall k,l \in R \quad and \quad u \in V, \quad (k+l)*u=k*u+l*u \] *notice: the two operation in a triple is not necessary to be add and multiply

Example:

Let $ V=R_{>0} $ be the set of strictly positive real numbers. Let \(u \boxplus v = u*v\) for all \(u,v \in V\) and \(k \odot u=u^k\) for all \(u \in V\) and \(k \in R\). Show that \((V, \boxplus, \odot)\) is a vector space.

Let \(u,v \in V\), then \(u \boxplus v = u*v\)
\(\because u\) and \(v\) both are positive real number
\(\therefore u * v\) is also a positive real number, so V is closure under \(\boxplus\).

Let \(v,1 \in V\), then \(v \boxplus 1 = v*1 = v\)
\(\therefore\) V has \(\boxplus\) identity.

Let \(v \in V\) and its \(\boxplus\) identity \(=1\), then \(v \boxplus (1/v)=v*(1/v)=1\),
\(\therefore\) V has \(\boxplus\) inverse.

Let \(u,v,x \in V\), then \(u \boxplus (v \boxplus x)=u \boxplus (v*x)=u*v*x=(u \boxplus v)*x=(u \boxplus v) \boxplus x\)
\(\therefore\) V has \(\boxplus\) associality.

Let \(u,v \in V\), then \(u \boxplus v=u*v=v*u=v \boxplus u\)
\(\therefore\) V has \(\boxplus\) commutativity.

Let \(v \in V\), \(k \in R\), then \(k \odot v = v^k\)
\(\because v\) is a positive real number and \(k\) is a real number
\(\therefore\) $v^k $ is also a positive real number, so V is closure under \(\odot\).

Let \(v \in V\), then \(1 \odot v=v^1=v\)
\(\therefore\) V has \(\odot\) identity.

Let \(v \in V\) and \(a,b \in R\), then \((b \odot a) \odot v=(v^a)^b=b \odot v^a=b \odot (a \odot v)\)
\(\therefore\) V has \(\odot\) associality.

Let \(u,v \in V\) and \(a,b \in R\),
then \(a \odot (u \boxplus v)=a \odot (u*v)=(u*v)^a=u^a * v^a=(a \odot u) \boxplus (a \odot v)\)
and \((a \boxplus b) \odot u=(a*b) \odot u=u^{a*b}=u^a * u^b=(a \odot u) \boxplus (b \odot u)\)
\(\therefore\) V has distributive property.

Therefore V fits all 9 rules and it's a vector space.