Orthobases

Dr. Matthieu R Bloch

Wednesday September 15, 2021

Logistics

Assigment 2
- Hard deadline on Thursday September 16, 2021 (hard)
Assignment 3
- Posted yesterday night
- Due Monday September 20, 2021 (soft)
- Hard deadline on Tuesday September 21, 2021 (exception!)
Midterm 1
- As of Wednesday, September 15, 2021 10:18am, Surveillance Incidence Positivity at 0.53%
- In-class exam during regular class time - 75 minutes
- Topics: everything up to and including today’s coverage
- Open notes (slides+HW solution + Prof. Romberg’s notes) - no electronics

Last time: Orthogonality principle
- Key take-way: Closest vector problem has nice properties: orthogonality principle
- This can be solve nicely computationally!
- Questions?
Today:
- Orthobases: some theory and more about infinite dimensional Hilbert spaces
- Examples!
Monday September 20, 2021: non orthogonal bases
Reading: Romberg, lecture notes 5 and 6

A collection of vectors \(\set{v_i}_{i=1}^n\) in a finite dimensional Hilbert space \(\calH\) is an orthobasis if 1) \(\text{span}(\set{v_i}_{i=1}^n)=\calH\); 2) \(\forall i\neq j\in\intseq{1}{n}\,v_i\perp v_j\); 3) \(\forall i\in\intseq{1}{n} \,\norm{v_i}=1\).
If the last condition is not met, this is just called an orthogonal basis
Orthobases are useful because we can write \(x=\sum_{i=1}^n\dotp{x}{v_i}v_i\) (what happens in a non-orthonormal basis?)
We would like to extend this idea to infinite dimension and happily write \(x=\sum_{i=1}^\infty\dotp{x}{v_i}v_i\)
- We have to be a bit careful
- With a little bit of machinery, this works: separability + completeness

Consider a Hilbert space \(\calH\). The following notions we encountered work without problems:
- Norm induced by inner product
- Cauchy-Schwartz inequality
- Orthogonality + Pythagorean theorem + projection theorem
A collection of vectors \(\set{v_i}_{i\in\calI}\) in a Hilbert space \(\calV\) is orthonormal if
1. \(\forall i\neq j\in\intseq{1}{n}\,v_i\perp v_j\);
2. \(\forall i\in\intseq{1}{n} \,\norm{v_i}=1\).
We do not even require \(\calI\) to be countable at this stage.
We are not saying anything about this being a basis

Let \(c\in\calH\) and let \(\set{x_n}_{n\geq 1}\) be a sequence of vectors in \(\calH\) that converges to \(x^*\in\calH\). Then \(\lim_{n\to\infty}\dotp{x_n}{c} = \dotp{x^*}{c}\)
Let \(\set{v_n}_{n\geq 1}\) be sequence of orthonormal vectors in \(\calH\). Then for any \(x\in\calH\), \(\sum_{n=1}^\infty\abs{\dotp{x}{v_n}}^2\leq\norm{x}^2\).
The following properties of an orthonormal set \(\set{e_i}_{i\geq 1}\) are equivalent:
1. Finite linear combinations of elements in \(\calH\) are dense in \(\calH\);
2. If \(v\in\calH\) and \(\forall j\geq 1\) \(\dotp{v}{e_j}=0\) then \(v=0\);
3. If \(v\in\calH\) then \(\sum_{i=1}^n\dotp{v}{e_i}e_i\) converges to \(v\) (in the norm \(\norm{\cdot}\)) as \(n\to\infty\);
4. If \(v\in\calH\) and \(\forall k\) \(a_k\eqdef \dotp{v}{e_k}\) then \(\norm{v}^2=\sum_{k\geq 1}\abs{a_k}^2\).
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
Key take away for separable Hilbert spaces
- \(x=\sum_{i=1}^\infty\dotp{x}{v_i}v_i\) is perfectly well defined for an orthonormal basis
- Parseval’s identity tell us that \(\norm{x}^2=\sum_{k\geq 1}\abs{\dotp{x}{v_k}}^2\).
- It looks like we don’t need to even worry about the nature of \(\calH\) and only think about coefficients \(\dotp{x}{v_i}\)
Any separable Hilbert space is isomorphic to \(\ell_2\)