Inner product is anti-linear in second argument

Dependencies:

  1. Inner product space
  2. /complex-numbers/conjugation-is-homomorphic

Let $V$ be an inner-product space. The inner product is anti-linear in the second argument. This means that:

Proof

\begin{align} & \langle \mathbf{u}, a \mathbf{v} \rangle \\ &= \overline{\langle a \mathbf{v}, \mathbf{u} \rangle} \tag{by conjugate symmetry} \\ &= \overline{a\langle \mathbf{v}, \mathbf{u} \rangle} \tag{by linearity in first argument} \\ &= \overline{a}\overline{\langle \mathbf{v}, \mathbf{u} \rangle} \tag{conjugation is homomorphic} \\ &= \overline{a} \langle \mathbf{u}, \mathbf{v} \rangle \tag{by conjugate symmetry} \end{align}

\begin{align} & \langle \mathbf{u}, \mathbf{v_1} + \mathbf{v_2} \rangle \\ &= \overline{\langle \mathbf{v_1} + \mathbf{v_2}, \mathbf{u} \rangle} \tag{by conjugate symmetry} \\ &= \overline{\langle \mathbf{v_1}, \mathbf{u} \rangle + \langle \mathbf{v_2}, \mathbf{u} \rangle} \tag{by linearity in first argument} \\ &= \overline{\langle \mathbf{v_1}, \mathbf{u} \rangle} + \overline{\langle \mathbf{v_2}, \mathbf{u} \rangle} \tag{conjugation is homomorphic} \\ &= \langle \mathbf{u}, \mathbf{v_1} \rangle + \langle \mathbf{u}, \mathbf{v_2} \rangle \tag{by conjugate symmetry} \end{align}

Dependency for:

  1. Extreme direction of convex cone as extreme point of intersection with hyperplane
  2. Symmetric operator iff hermitian
  3. Symmetric operator on V has a basis of orthonormal eigenvectors
  4. All eigenvalues of a symmetric operator are real
  5. x and y are parallel iff ∥x∥²∥y∥² = |< x, y >|².
  6. Pythagorean theorem
  7. Cauchy-Schwarz Inequality
  8. Joining orthogonal linindep sets

Info:

Transitive dependencies:

  1. /complex-numbers/conjugation-is-homomorphic
  2. Group
  3. Ring
  4. Field
  5. Vector Space
  6. Inner product space