Non-orthogonal bases
Dr. Matthieu R Bloch
Monday, September 27, 2021
Logistics
- Assignments 2/3: Grading ongoing
- Midterm 1
- Grading ongoing
- General logistics of exams
- Assignment 4 assigned today or tomorrow
What’s on the agenda for today?
Last time: Orthobases
- Things work in infinite dimension too with orthobases (without thinking too much)
- Last little bit to prove: isomorphism to
Today: What happens with non-orthogonal bases
Wednesday September 29, 2021: Least-square regression
Reading: Romberg, lecture notes 7
Recap - Orthobases in infinite dimension
The following properties of an orthonormal set are equivalent:
- Finite linear combinations of elements in are dense in ;
- If and then ;
- If then converges to (in the norm ) as ;
- If and then .
This does not say anything about the existence of such nice orthornormal set
A space is separable if it contains a countable dense subset.
Separability is the key property to deal with sequences instead of collections
Any separable Hilbert space has an orthonormal basis.
Most useful Hilbert spaces are separable! We won’t worry about non-separable Hilbert spaces
Any separable Hilbert space is isomorphic to
Non-orthogonal bases in finite dimension
Let be a linearly independent set in a Hilbert space of dimension . Then, for any , for some . In addition, there exists such that
Inequality is tight on both sides
For orthobases,
Interpretation:
- The values of and govern the stability of the representation
Examples
Non-orthogonal bases in finite dimension: dual basis
Non-orthogonal bases in infinite dimension
is a Riesz basis for Hilbert space if and there exists such that uniformly for all sequences with .
In infinite dimension, the existence of is not automatic.
Examples
Non-orthogonal bases in finite dimension: dual basis
Computing expansion on Riesz basis not as simple in infinite dimension: Gram matrix is “infinite”
The Grammiam is a linear operator
Fact: there exists another linear operator such that
We can replicate what we did in finite dimension.


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Non-orthogonal bases
Dr. Matthieu R Bloch
Monday, September 27, 2021