1. Inner product spaces: general Ideas such that Energy: Use inner product to measure correlation : well-correlated: uncorrelated:, Length: Recall definition:inner product space when

2. , spaces: Index set : finite or infinite discrete case all sequences: with finite energy: Inner product: Examples:, finite means

3. , spaces: Continuous case Suppose integrals defined on all functions with finite energy: Inner product: Examples:

4. Inner product spaces: general properties Cauchy-Schwartz Inequality: Triangle Inequality: Polarization:

5. Inner product spaces: general Ideas Remember: can now be infinite-dimensional. So we have to take more care! Family in orthonormal when Bessel’s Inequality:

6. Inner product spaces: general Ideas Complete orthonormal family: When complete: in sense Parseval:

7. Examples orthonormal families: 1. 2. 3. 4. standard basis, Fourier basis complete in Haar type complete in translates of box function in. Complete? NO translates and dilates of box function in. Complete? NO

8. More Examples: 5.translates of Haar function in. Complete? NO Prototypical wavelet idea: 6. splitting translates of box and Haar function in Complete? NO. But complete in

9. Operators on inner product spaces In infinite dimensions need to worry about controlling energy:bounded when Energy-preserving when Basic example: Synthesizing

10. Basic example Adjoint: Analyzing Synthesizing orthonormal in Check energy: Bessel’s inequality