section_path: Duality/Inner-product spaces
section_name: Inner-product spaces
paragraph_text: Inner-product spaces
  
  Besides these basic concepts, linear algebra also studies vector spaces with additional structure, such as an inner product. The inner product is an example of a bilinear form, and it gives the vector space a geometric structure by allowing for the definition of length and angles. Formally, an inner product is a map
  
  that satisfies the following three axioms for all vectors u, v, w in V and all scalars a in F:
  
  We can define the length of a vector v in V by
  
  and we can prove the Cauchy–Schwarz inequality:
  
  In particular, the quantity
  
  and so we can call this quantity the cosine of the angle between the two vectors.
  
  Two vectors are orthogonal if ⟨ u, v ⟩ = 0. An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other. Given any finite-dimensional vector space, an orthonormal basis could be found by the Gram–Schmidt procedure. Orthonormal bases are particularly easy to deal with, since if v = a v + ⋯ + a v, then
  
  The inner product facilitates the construction of many useful concepts. For instance, given a transform T, we can define its Hermitian conjugate T* as the linear transform satisfying
  
  If T satisfies TT* = T*T, we call T normal. It turns out that normal matrices are precisely the matrices that have an orthonormal system of eigenvectors that span V.