Inner Product

Last updated Nov 1, 2022

Let $V$ be a Vector Space over $F$, which is either the Real Numbers or Complex Numbers. Then an Inner Product on $V$ is a Function $\langle \cdot, \cdot \rangle: V \times V \to F$ that satisfies the following properties

1. Conjugate Symmetry: $\langle \mathbf{u}, \mathbf{v} \rangle = \overline{\langle \mathbf{v}, \mathbf{u} \rangle}$ for all $\mathbf{u}, \mathbf{v} \in V$.
2. Inner Products are Linear in their first argument: $\langle a \mathbf{u} + b \mathbf{v}, \mathbf{w} \rangle = a \langle \mathbf{u}, \mathbf{w} \rangle + b \langle \mathbf{v}, \mathbf{w} \rangle$
3. Positive Definiteness: $\langle \mathbf{u}, \mathbf{u} \rangle \geq 0$ with equality iff $\mathbf{u} = \mathbf{0}$. The $(\Leftarrow)$ direction is implied by the other properties.

TODO

• Complex Conjugate