Inner%20product%20spaces:%20general%20Ideas

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Inner product spaces general Ideas
such that
Energy
, Length
Use inner product to measure correlation
well-correlated
uncorrelated
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, spaces
discrete case
Index set finite or infinite
all sequences
with finite energy
Inner product
finite means
Examples
,
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, spaces
Continuous case
Suppose integrals defined on
all functions
with finite energy
Inner product
Examples
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Inner product spaces general properties
Cauchy-Schwartz Inequality
Triangle Inequality
Polarization
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Inner product spaces general Ideas
Remember can now be infinite-dimensional.
So we have to take more care!
Family in
orthonormal when
Bessels Inequality
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Inner product spaces general Ideas
Complete orthonormal family
When complete
in sense
Parseval
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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
. Complete? NO
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More Examples
5.
translates

of Haar function in
. Complete? NO
Prototypical wavelet idea
splitting
6.
translates
of box and Haar function in
.
Complete? NO
But complete in
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Operators on inner product spaces
In infinite dimensions need to worry about
controlling energy
bounded when
Energy-preserving when
Basic example Synthesizing
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Basic example
Synthesizing
orthonormal in
Check energy
Adjoint Analyzing
Bessels inequality