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Code: | M3003 | Acronym: | M3003 | Level: | 300 |

Keywords | |
---|---|

Classification | Keyword |

OFICIAL | Mathematics |

Active? | Yes |

Responsible unit: | Department of Mathematics |

Course/CS Responsible: | First Degree in Mathematics |

Acronym | No. of Students | Study Plan | Curricular Years | Credits UCN | Credits ECTS | Contact hours | Total Time |
---|---|---|---|---|---|---|---|

L:B | 0 | Official Study Plan | 3 | - | 6 | 56 | 162 |

L:CC | 0 | Plano de estudos a partir de 2014 | 2 | - | 6 | 56 | 162 |

3 | |||||||

L:F | 0 | Official Study Plan | 2 | - | 6 | 56 | 162 |

3 | |||||||

L:G | 0 | study plan from 2017/18 | 3 | - | 6 | 56 | 162 |

L:M | 10 | Official Study Plan | 2 | - | 6 | 56 | 162 |

3 | |||||||

L:Q | 0 | study plan from 2016/17 | 3 | - | 6 | 56 | 162 |

Teacher | Responsibility |
---|---|

Carlos Miguel de Menezes |

Theoretical and practical : | 4,00 |

Type | Teacher | Classes | Hour |
---|---|---|---|

Theoretical and practical | Totals | 1 | 4,00 |

Carlos Miguel de Menezes | 4,00 |

Last updated on 2018-10-22.

Fields changed: Learning outcomes and competences, Palavras Chave, Componentes de Avaliação e Ocupação, Programa

Fields changed: Learning outcomes and competences, Palavras Chave, Componentes de Avaliação e Ocupação, Programa

The student is supposed to acquire during the course proficiency with the main concepts and theorems of Functional Analysis in Banach spaces and Hilbert spaces, with applications to classical function spaces.

1.The student is supposed to acquire during the course proficiency with the main concepts and theorems of Functional Analysis in Banach spaces and Hilbert spaces, with applications to classical function spaces:

a. In Banach spaces: norms, equivalente norms, space of bounded linear maps, dual and bidual space, reflexive spaces, weak topologies, the so called four pilars of Functional Analysis: theorems of Hahn-Banach, open mapping, closed graph and uniform boundedness principle, and weak topologies. Fixed Point theorems of Banach, Brouwer and Schauder.

b. In Hilbert spaces: orthogonality, Riesz representation theorem, separability, orthonormal families and bases, adjoint operator, projections and idempotents, compact operators, spectral theory of compact normal operators

2. Applications of these results to classical function spaces: sequence spaces, continuous functions spaces, and integrable functions, and, if time allows, application of spectral theory of compact self adjoint operators to Sturm-Liouville systems. Several examples of rlevant operators and functional spaces will be the subject of problems/exercises to be solved by the student in practice lectures or autonomus at home.

Real Analysis/Calculus in one and several variables, Complex Analysis, basic notions on metric spaces aquired in real analysis courses, Linear Algebra in finite dimensions.

Have attended an introdcutory course in topology/metric spaces will be a great help.

I. Real and Complex Normed spaces and Banach spaces:

1. General concepts: norm, associated distance, open and closed balls. Sequences and series in a normed space. Cauchy sequences and fast Cauchy sequences. Banach spaces. Bounded linear maps. The normed space of bounded linear maps between two normed spaces. Banach fixed point theorem.

2. Equivalent norms. All norms in a finite dimensional vector space are equivalent. locally compact normed spaces are the finite dimensional normed spaces. Sums and products of normed spaces, norm in the quotient of a normed space by a closed subspace.

3. Examples of function spaces with the norm of uniform convergence:

bounded functions, bounded continuous functions, continuous functions with compact support, continuous function sthat vanish at infinity. Banach Spaces of $C^k$ functions. Study of completeness of the above spaces.

Sequence spaces related to previous spaces as particular cases.

4. Dini theorem for monotone sequence of continuous functions defined in a compact space and with real values. Stone-Weierstrass theorem.

5. Equicontinuity and Ascoli-Arzéla theorem.

6. Given a bounded continuous map defined in a dense subspace of a Banach space X with values in another Banach space Y, extension of that map to the Banach space X. Completion of a normed space.

7. Baire category theorem. Open mapping theorem. Banach isomorphism theorem.

8. Closed graph theorem. Uniform boundedness principle. Banach-Steinhaus theorem.

9. Brouwer and Schauder fixed point theorems.

10. Topological dual and bidual of a normed space. Hahn-Banach theorem in analytic and geometric form. Reflexive Banach spaces. Transpose (or adjoint) of a bounded linear map. Closed range theorem.

11. Weak and weak^*-convergence in a Banach space. Banach-Alaoglu theorem.

12. Young, Holder and Minkowski inequalities (discrete and contínuous).

Space of Lp-Riemnan integrable functions in compacts of R^d. Proof that for these normed spaces are not complete (1<=p< \infty)

13. Lebesgue integral in euclidean spaces as extension of Riemann integral to the completion in L^1 norm of the space of continous functions with compact support.

II. Espaços de Hilbert:

1. General concepts: Sesquilinear forms, inner products.

Polarization identity, Cauchy-Schwarz inequality, norm associated to the inner product, paralelogram law, orthogonal vectors, Pitagoras theorem.

Prehilbertian and Hilbert spaces.

Orthogonality and biorthogonality of vector subspaces.

2. Orthogonal sum of Hilbert spaces. Orthogonal projection

on a convex closed set and associated orhogonal decomposition of the Hilbert space. Riez representation theorem of the topologiacl dual of a Hilbert space. Completion of a pre-Hilbert space. Weak topology in a Hilbert space.

3. Total family, orthonormal family, and orthonormal basis of a

Hilbert space. Extension of orthonormal family to a orthonormal basis.

Dimension of the Hilbert space. Separable Hilbert spaces.

Gram-Schmidt orthonormalization of total famíly. Bessel inequality, Parseval identity, Fourier series in Hilbert spaces. Isometric Hilbert spaces.

4. Adjoint of a bounded inear operator between Hilbert spaces and its main proprerties. Self adjoint and normal operators. Characterization of bounded invertible linear operators in terms of the adjoint.

5. Projections, idempotents, unitary operators, partial isometries, invariant and resctriction subspaces.

6. Compact operators in Hilbert spaces.

7. Spectral theory of compact self adjoint and normal operators.

Pedersen, Gert; Analysis Now, Springer

Presencial lectures on the contents of the syllabus given by the teacher.

The students will work in classes and at home on exercises and problems proposed by the teacher.

designation | Weight (%) |
---|---|

Exame | 100,00 |

Total: |
100,00 |

designation | Time (hours) |
---|---|

Estudo autónomo | 106,00 |

Frequência das aulas | 56,00 |

Total: |
162,00 |

Any type or special examination can be from one the following types: exclusively by an oral examination, only a written exam, one oral examination and a written exam.

The decision of which of the above types is each special examination is exclusively the responsability of the teacher assigned to the curricular unit.

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Page created on: 2022-12-08 at 06:18:21 | Reports Portal

Page created on: 2022-12-08 at 06:18:21 | Reports Portal