Let V {\displaystyle {}V} be a K {\displaystyle {}{\mathbb {K} }} -vector space. An inner product on V {\displaystyle {}V} is a mapping
satisfying the following properties:
for all λ 1 , λ 2 ∈ K {\displaystyle {}\lambda _{1},\lambda _{2}\in {\mathbb {K} }} , x 1 , x 2 , y ∈ V {\displaystyle {}x_{1},x_{2},y\in V} , and
for all λ 1 , λ 2 ∈ K {\displaystyle {}\lambda _{1},\lambda _{2}\in {\mathbb {K} }} , x , y 1 , y 2 ∈ V {\displaystyle {}x,y_{1},y_{2}\in V} .
for all v , w ∈ V {\displaystyle {}v,w\in V} .