Hilbert Spaces
Inner Product Space
Hilbert Space
Examples
Example 1: with
Example 2: with
Example 3: with
Example 4: (square-integrable functions on )
Norm from Inner Product
Parallelogram Law
A norm comes from an inner product if and only if it satisfies the parallelogram law.
Orthogonality
Pythagorean Theorem
Orthogonal Projection
The orthogonal projection onto is .
Riesz Representation Theorem
Orthonormal Systems
Bessel's Inequality
Parseval's Identity
Orthonormal Basis
Gram-Schmidt Process
Separable Hilbert Spaces
Applications
- Quantum mechanics (state spaces)
- Signal processing (Fourier analysis)
- Statistics (reproducing kernel Hilbert spaces)
- Machine learning (kernel methods)
- Partial differential equations (weak solutions)