inner product space

An inner product space is a vector space endowed with an inner product.

Formally,
An inner product space consists of a vector space $V$ over field $F$ equipped with the inner product $⟨\cdot ,\cdot ⟩:V\times V\to F$, such that for any vectors $x,y,z\in V$ and scalars $a,b\in F$,

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