Inner Product

The Inner Product is a Bilinear map $⟨\cdot ,\cdot ⟩:V\times V\to F$ satisfying:

• Symmetry: $⟨\vec{a},\vec{b}⟩=⟨\vec{b},\vec{a}⟩$
• Linearity: $⟨\alpha \vec{a}+\beta \vec{b},\vec{c}⟩=\alpha ⟨\vec{a},\vec{c}⟩+\beta ⟨\vec{b},\vec{c}⟩$
• Positive definiteness: $⟨\vec{a},\vec{a}⟩\ge 0$, with equality iff $\vec{a}=0$

The standard inner product on ${\mathbb{R}}^n$ is the Dot Product:

$⟨\vec{a},\vec{b}⟩={\sum }_i{a}_i{b}_i=|\vec{a}||\vec{b}|\cos \theta$