Inner Product

Definition

An inner product on a vector space $V$ over $\mathbb{R}$ or $\mathbb{C}$ is a function $\langle \cdot, \cdot \rangle: V \times V \to \mathbb{C}$ (or $\mathbb{R}$) satisfying:

1. Conjugate symmetry: $\langle x, y \rangle = \overline{\langle y, x \rangle}$
2. Linearity in first argument: $\langle ax + by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle$
3. Positive definiteness: $\langle x, x \rangle \geq 0$, with equality iff $x = 0$

For Functions

On function spaces, the standard inner product is:

$$ \langle f, g \rangle = \int f(x) \overline{g(x)} \, dx $$

This generalizes the dot product from finite-dimensional vectors to infinite-dimensional function spaces.

Intuition

The inner product measures the “similarity” or “overlap” between two vectors or functions:

• $\langle f, g \rangle = 0$ means $f$ and $g$ are orthogonal (perpendicular)
• $|\langle f, g \rangle|$ large means $f$ and $g$ are similar

Related Concepts

• norm — defined via $\|f\| = \sqrt{\langle f, f \rangle}$
• L² space — functions with finite $\langle f, f \rangle$
• Orthogonality — when $\langle f, g \rangle = 0$