Inner Product Spaces
Definition
Norm Induced by Inner Product
Cauchy-Schwarz Inequality
Equality holds if and only if and are linearly dependent.
Orthogonality
Orthogonal Set
A set is orthogonal if for .
Orthonormal Set
A set is orthonormal if (Kronecker delta).
Inner Product Spaces Examples
Example 1: with .
Example 2: with .
Example 3: with weighted inner product ( positive definite).
Orthogonal Projection
Pythagorean Theorem
Orthogonal Complement
Best Approximation
Gram Matrix
For a basis , the Gram matrix is .
The basis is linearly independent if and only if .
Applications
- Least squares approximation
- Fourier series (orthogonal basis of functions)
- Quantum mechanics (Hilbert spaces)
- Machine learning (kernel methods)
- Signal processing