Inner Product Spaces
Inner products generalize the dot product, allowing notions of length, angle, orthogonality, and projection. They are essential for Hilbert spaces and optimization.
Key Points
- An inner product is linear, symmetric (or conjugate symmetric), and positive definite.
- Orthogonality and orthonormal bases simplify many computations.
- The projection onto a subspace minimizes the distance to that subspace.
Formulas
Inner product axioms
$$\langle u, v \rangle = \overline{\langle v, u \rangle}, \quad \langle u, u \rangle \ge 0$$
Cauchy-Schwarz
$$|\langle u, v \rangle| \le \|u\| \|v\|$$
Projection
$$\operatorname{proj}_v u = \frac{\langle u, v \rangle}{\langle v, v \rangle} v$$
Code Example
import numpy as np
u = np.array([1, 2, 3])
v = np.array([1, 0, 1])
proj = (np.dot(u, v) / np.dot(v, v)) * v
print(proj)