section_path: Dot product and generalisations/Inner product
section_name: Inner product
paragraph_text: Inner product
  
  To define angles in an abstract real inner product space, we replace the Euclidean dot product (·) by the inner product ⟨ ⋅, ⋅ ⟩ {\displaystyle \langle \cdot,\cdot \rangle }, i.e.
  
  ⟨ u, v ⟩ = cos (θ) ‖ u ‖ ‖ v ‖. {\displaystyle \langle \mathbf {u},\mathbf {v} \rangle =\cos(\theta)\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
  
  In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with
  
  Re (⟨ u, v ⟩) = cos (θ) ‖ u ‖ ‖ v ‖. {\displaystyle \operatorname {Re} \left(\langle \mathbf {u},\mathbf {v} \rangle \right)=\cos(\theta)\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
  
  or, more commonly, using the absolute value, with
  
  | ⟨ u, v ⟩ | = | cos (θ) | ‖ u ‖ ‖ v ‖. {\displaystyle \left|\langle \mathbf {u},\mathbf {v} \rangle \right|=\left|\cos(\theta)\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
  
  The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces span (u) {\displaystyle \operatorname {span} (\mathbf {u})} and span (v) {\displaystyle \operatorname {span} (\mathbf {v})} spanned by the vectors u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } correspondingly.