Inner product spaces general Ideas Recall definition when

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Inner product spaces: general Ideas Recall definition: when inner product space such that Energy:

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

, spaces: discrete case Index set : finite or infinite all sequences: with finite

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

, spaces: Continuous case Suppose integrals defined on all functions with finite energy: Inner

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

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

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

Inner product spaces: general Ideas Remember: can now be infinite-dimensional. So we have to

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

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

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

Examples orthonormal families: 1. standard basis, Fourier basis complete in 2. Haar type complete

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

More Examples: 5. translates of Haar function in . Complete? Prototypical wavelet idea: splitting

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

Operators on inner product spaces In infinite dimensions need to worry about controlling energy:

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

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

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

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