Orthogonality principle and Orthobases

Logistics

Assigment 2
- Due Tuesday September 14, 2021 (soft)
- Due Thursday September 16, 2021 (hard)
Assignment 3
- Posted Tuesday September 14, 2021
- Due Monday September 20, 2021 (soft)
Midterm 1
- Moved to take home exam if Surveillance Incidence Positivity >1
- https://health.gatech.edu/coronavirus/health-alerts
- As of Sunday, September 12, 2021 2:45pm: 1.02%

Last time: Hilbert spaces
- Key take-way: we can avoid problems in infinite dimensions
- Key concepts: completeness is the property that we want
- Questions?
Today:
- Orthogonality principle: computing closest points
- Orthobases: some theory and more about infinite dimensional Hilbert spaces
Wednesday September 15, 2021: lots of examples
Reading: Romberg, lecture notes 5 and 6

Let $H$ be a Hilbert space with induced norm ${⟨ \cdot, \cdot ⟩}_{}$ and induced norm ${‖ \cdot ‖}_{}$ ; let $T$ be subspace of $H$
For $x \in H$ , what is the closest point of $\hat{x} \in T$ ? How do we solve $min_{y \in T} {‖ x - y ‖}_{}$ ?
This problem has a unique solution given by the orthogonality principle
Let $X$ be a pre-Hilbert space, $T$ be a subspace of $X$ , and $x \in X$ .

If there exists a vector $m^{*} \in T$ such that $\forall m \in T$ ${‖ x - m^{*} ‖}_{} \leq {‖ x - m ‖}_{}$ , then $m^{*}$ is unique.
$m^{*} \in T$ is a unique minimizer if and only if the error $x - m^{*}$ be orthogonal to $T$ .
This doesn’t say that $m^{*}$ exists!
Let $H$ be a Hilbert space, $T$ be a closed subspace of $X$ , and $x \in X$ .

There exists a unique vector $m^{*} \in T$ such that $\forall m \in T$ ${‖ x - m^{*} ‖}_{} \leq {‖ x - m ‖}_{}$ .
$m^{*} \in T$ is a unique minimizer if and only if the error $x - m^{*}$ be orthogonal to $T$

The orthogonality principle gives us a procedure for computing the closest point
Let $H$ be a Hilbert space, $T$ be a subspace of $H$ with dimension $n$ , and $x \in H$ . Let ${e_{i}}_{i = 1}^{n}$ be a basis for $T$ . Then the projection $\hat{x}$ of $x$ onto $T$ is $\hat{x} = \sum_{i = 1}^{n} α_{i} e_{i}$ where $α ≜ {[\begin{matrix} α_{1} & \dots & α_{n} \end{matrix}]}^{⊺}$ is the solution of $G α = b$ with $G$ the Grammiam matrix of the basis and $b$ the coordinates of $x$ on the basis.