Orthobases

Logistics

Assigments 1&2: solution posted for #1, coming today for #2
Assignment 3
- Hard deadline on Tuesday September 21, 2021 (exception!)
- This is what matters most Midterm
Midterm 1
- 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 - no textbooks
- Prof. Bloch office hours: Tuesday September 21, 2021 8am on https://bluejeans.com/9058052289

Last time: More on orthobases
- Orthobases: things work in infinite dimension too
- Examples!
Monday September 27, 2021: non orthogonal bases
Reading: Romberg, lecture notes 5 and 6

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

Consider a Hilbert space $H$ . 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 ${v_{i}}_{i \in I}$ in a Hilbert space $V$ is orthonormal if
1. $\forall i \neq j \in [1; n] v_{i} ⊥ v_{j}$ ;
2. $\forall i \in [1; n] {‖ v_{i} ‖}_{} = 1$ .
We do not even require $I$ to be countable at this stage.
We are not saying anything about this being a basis

Let $c \in H$ and let ${x_{n}}_{n \geq 1}$ be a sequence of vectors in $H$ that converges to $x^{*} \in H$ . Then $lim_{n \to \infty} {⟨ x_{n}, c ⟩}_{} = {⟨ x^{*}, c ⟩}_{}$
Let ${v_{n}}_{n \geq 1}$ be sequence of orthonormal vectors in $H$ . Then for any $x \in H$ , $\sum_{n = 1}^{\infty} {| {⟨ x, v_{n} ⟩}_{} |}^{2} \leq {‖ x ‖}_{}^{2}$ .
The following properties of an orthonormal set ${e_{i}}_{i \geq 1}$ are equivalent:
1. Finite linear combinations of elements in $H$ are dense in $H$ ;
2. If $v \in H$ and $\forall j \geq 1$ ${⟨ v, e_{j} ⟩}_{} = 0$ then $v = 0$ ;
3. If $v \in H$ then $\sum_{i = 1}^{n} {⟨ v, e_{i} ⟩}_{} e_{i}$ converges to $v$ (in the norm ${‖ \cdot ‖}_{}$ ) as $n \to \infty$ ;
4. If $v \in H$ and $\forall k$ $a_{k} ≜ {⟨ v, e_{k} ⟩}_{}$ then ${‖ v ‖}_{}^{2} = \sum_{k \geq 1} {| a_{k} |}^{2}$ .
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
- $x = \sum_{i = 1}^{\infty} {⟨ x, v_{i} ⟩}_{} v_{i}$ is perfectly well defined for an orthonormal basis
- Parseval’s identity tell us that ${‖ x ‖}_{}^{2} = \sum_{k \geq 1} {| {⟨ x, v_{k} ⟩}_{} |}^{2}$ .
- It looks like we don’t need to even worry about the nature of $H$ and only think about coefficients ${⟨ x, v_{i} ⟩}_{}$
Any separable Hilbert space is isomorphic to $ℓ_{2}$