Real Inner Product Space

Definition

An inner product on a real vector space $V$ is a function that associates a real number $⟨\mathbf{u},\mathbf{v}⟩$ with each pair of vectors in $V$ in such a way that the following axioms are satisfied for all vectors $\mathbf{u},\mathbf{v},\mathbf{w}\in V$ and all scalars $k$.

1. $⟨\mathbf{u},\mathbf{v}⟩=⟨\mathbf{v},\mathbf{u}⟩$ (Symmetry axiom)
2. $⟨\mathbf{u}+\mathbf{v},\mathbf{w}⟩=⟨\mathbf{u},\mathbf{w}⟩+⟨\mathbf{v},\mathbf{w}⟩$ (Additivity axiom)
3. $⟨k\mathbf{u},\mathbf{v}⟩=k⟨\mathbf{u},\mathbf{v}⟩$ (Homogeneity axiom)
4. $⟨\mathbf{v},\mathbf{v}⟩\ge 0$ and $⟨\mathbf{v},\mathbf{v}⟩=0$ if and only if $\mathbf{v}=v{f}_0$ (Positivity axiom)

A real vector space with an inner product is called a real inner product space