Inner product spaces: general Ideas

such that

Energy:Use inner product to measure correlation :

well-correlated:

uncorrelated:

, Length:

Recall definition: inner product spacewhen

, spaces:

Index set : finite or infinite

discrete case

all sequences:

with finite energy:

Inner product:

Examples:

,

finite means

, 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 Ideas

Remember: can now be infinite-dimensional.

So we have to take more care! Family in

orthonormal when

Bessel’s Inequality:

Inner product spaces: general Ideas

Complete orthonormal family:

When complete:

in sense

Parseval:

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

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

Operators on inner product spaces

In infinite dimensions need to worry about

controlling energy: bounded when

Energy-preserving when

Basic example: Synthesizing

Basic example

Adjoint: Analyzing

Synthesizing

orthonormal in

Check energy:

Bessel’s inequality