# Course Details for A.Y. 2016/2017

#### Name:

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

### Basic information

##### Credits:

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

##### Degree(s):

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

##### 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. ArzelĂ '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.
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- H. Brezis, Funtional Analysis, Sobolev Spaces, and partial differential equations , Springer.
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### Course page updates

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

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 30 settembre 2016, 16:37*