Cauchy-Schwarz Inequality

Dependencies:

  1. Inner product space
  2. Zero in inner product
  3. Inner product is anti-linear in second argument

Let $V$ be an inner product space and $\mathbf{u}, \mathbf{v} \in V$. Then $|\langle \mathbf{u}, \mathbf{v} \rangle|^2 \le \langle \mathbf{u}, \mathbf{u} \rangle \langle \mathbf{v}, \mathbf{v} \rangle$.

Proof

\[ \mathbf{v} = 0 \Rightarrow |\langle \mathbf{u}, \mathbf{v} \rangle|^2 = 0 = \langle \mathbf{u}, \mathbf{u} \rangle \langle \mathbf{v}, \mathbf{v} \rangle \]

\[ k = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{v}\|^2} \Rightarrow \langle \mathbf{u}, \mathbf{v} \rangle = k\|\mathbf{v}\|^2 \]

\begin{align} 0 &\le \| \mathbf{u} - k\mathbf{v} \|^2 \tag{by positive semidefiniteness} \\ &= \langle \mathbf{u} - k\mathbf{v}, \mathbf{u} - k\mathbf{v} \rangle \\ &= \langle \mathbf{u}, \mathbf{u} \rangle - \langle \mathbf{u}, k\mathbf{v} \rangle - \langle k\mathbf{v}, \mathbf{u} \rangle + \langle k\mathbf{v}, k\mathbf{v} \rangle \tag{by (anti-)linearity} \\ &= \langle \mathbf{u}, \mathbf{u} \rangle - \overline{k}\langle \mathbf{u}, \mathbf{v} \rangle - k\overline{\langle \mathbf{u}, \mathbf{v} \rangle} + k\overline{k}\langle \mathbf{v}, \mathbf{v} \rangle \tag{by (anti-)linearity and conjugate symmetry} \\ &= \|\mathbf{u}\|^2 - \overline{k}k\|\mathbf{v}\|^2 - k\overline{k\|\mathbf{v}\|^2} + k\overline{k}\|\mathbf{v}\|^2 \\ &= \|\mathbf{u}\|^2 - k\overline{k}\|\mathbf{v}\|^2 - k\overline{k}\|\mathbf{v}\|^2 + k\overline{k}\|\mathbf{v}\|^2 \\ &= \|\mathbf{u}\|^2 - k\overline{k}\|\mathbf{v}\|^2 \\ &= \|\mathbf{u}\|^2 - \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{v}\|^2} \frac{\overline{\langle \mathbf{u}, \mathbf{v} \rangle}}{\|\mathbf{v}\|^2} \|\mathbf{v}\|^2 \\ &= \|\mathbf{u}\|^2 - \frac{|\langle \mathbf{u}, \mathbf{v} \rangle|^2}{\|\mathbf{v}\|^2} \end{align}

Therefore, $|\langle \mathbf{u}, \mathbf{v} \rangle|^2 \le \|\mathbf{u}\|^2\|\mathbf{v}\|^2$

Dependency for:

  1. Cauchy-Schwarz inequality for random variables
  2. Triangle inequality

Info:

Transitive dependencies:

  1. /complex-numbers/conjugation-is-homomorphic
  2. Group
  3. Ring
  4. Field
  5. Vector Space
  6. Inner product space
  7. Inner product is anti-linear in second argument
  8. Zero in inner product