Dr. Matthieu R Bloch

Monday September 13, 2021

**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 \(\calH\) be a Hilbert space with induced norm \(\dotp{\cdot}{\cdot}\) and induced norm \(\norm{\cdot}\) ; let \(\calT\) be subspace of \(\calH\)

For \(x\in\calH\), what is the closest point of \(\hat{x}\in\calT\)? How do we solve \(\min_{y\in\calT}\norm{x-y}\)?

This problem has a unique solution given by the orthogonality principle

Let \(\calX\) be a

*pre-Hilbert*space, \(\calT\) be a subspace of \(\calX\), and \(x\in\calX\).If there exists a vector \(m^*\in\calT\) such that \(\forall m\in\calT\) \(\norm{x-m^*}\leq \norm{x-m}\), then \(m^*\) is unique.

\(m^*\in\calT\) is a unique minimizer if and only if the error \(x-m^*\) be orthogonal to \(\calT\).This doesn’t say that \(m^*\) exists!

Let \(\calH\) be a

*Hilbert*space, \(\calT\) be a*closed*subspace of \(\calX\), and \(x\in\calX\).There exists a unique vector \(m^*\in\calT\) such that \(\forall m\in\calT\) \(\norm{x-m^*}\leq \norm{x-m}\).

\(m^*\in\calT\) is a unique minimizer if and only if the error \(x-m^*\) be orthogonal to \(\calT\)

The orthogonality principle gives us a procedure for computing the closest point

- Let \(\calH\) be a
*Hilbert*space, \(\calT\) be a subspace of \(\calH\) with dimension \(n\), and \(x\in\calH\). Let \(\set{e_i}_{i=1}^n\) be a basis for \(\calT\). Then the projection \(\hat{x}\) of \(x\) onto \(\calT\) is \[ \hat{x} = \sum_{i=1}^n\alpha_i e_i \] where \(\bfalpha\eqdef\mat{c}{\alpha_1&\cdots&\alpha_n}^\intercal\) is the solution of \(\bfG\bfalpha=\bfb\) with \(\bfG\) the Grammiam matrix of the basis and \(\bfb\) the coordinates of \(x\) on the basis.