inner product

An inner product is an operation on two vectors in a vector space which produces a scalar. In Euclidean geometry, the inner product is the dot product (scalar product) of Cartesian coordinates.

Formally,
an inner product on a vector space $V$ over field $F$ is the operation $⟨\cdot ,\cdot ⟩:V\times V\to F$ such that for any vectors $x,y,z\in V$ and scalars $a,b\in F$,

• $⟨x,y⟩=\overline{⟨y,x⟩}$ (conjugate symmetry)
• $⟨ax+by,z⟩=a⟨x,z⟩+b⟨y,z⟩$ (linearity)
• $⟨x,x⟩>0$ for $x\ne \mathbf{0}$ (positive definiteness)
  where $\bar{z}$ is the complex conjugate of $z$.

For any vectors $x,y,z\in V$ and scalars $a,b\in F$,

• $⟨\mathbf{0},x⟩=⟨x,\mathbf{0}⟩=0$
• $⟨x,x⟩$ is real and nonnegative
• $⟨x,x⟩=0$ if and only if $x=\mathbf{0}$
• $⟨x,ay+bz⟩=\bar{a}⟨x,y⟩+\bar{b}⟨x,z⟩$
• $⟨x+y,x+y⟩=⟨x,x⟩+2\cdot \mathrm{\Re }(⟨x,y⟩)+⟨y,y⟩$
  where $\mathrm{\Re }$ returns the real part of its argument.