# Course Details

#### Name:

**Functional Analysis In Applied Mathematics And Engineering / Functional Analysis In Applied Mathematics And Engineering**

### Basic information

##### Credits:

*Master Degree in Mathematical Engineering:* 9 Ects (b)

##### Degree(s):

Compulsory 1^{st} year Master Degree in Mathematical Engineering curriculum Comune

##### Language:

English

### Course Content

- Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
- Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
- Crash course on Lebesgue meausre and integration. Limit exchange theorema. Lp spaces. Completeness of Lp spaces.
- Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
- Hilbert spaces. Elemtary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
- Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
- Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.
- Introduction to the theory of unbounded operators. Linear differential operators. Applications.
- Introduction to infinite-dimensional differential calculus and variational methods

### Textbooks

- John K. Hunter, Bruno Nachtergaele,
**Applied Analysis**. World Scientific. * *
- H. Brezis,
**Funtional Analysis, Sobolev Spaces, and partial differential equations**. Springer. * *

### Online Teaching Resources

### Course page updates

This course page is available (with possible updates) also for the following academic years:

*Course information last updated on: 30 settembre 2016, 16:37*