Introduction to Topology

Code:

M3019

Acronym:

M3019

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2022/2023 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Computer Science

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	1	Official Study Plan	3	-	6	56	162
L:CC	1	study plan from 2021/22	3	-	6	56	162
L:F	1	Official Study Plan	2	-	6	56	162
			3
L:G	0	study plan from 2017/18	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162

Teaching language

Portuguese

Objectives

Understanding of certain classical theorems of topology and functional analysis, and their applications to mathematical analysis.

Learning outcomes and competences

Described in the objectives.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Real Analysis I, II, III
Linear Algebra and Analytical Geometry I, II

Program

I - Elements of Topology:

• Toplogical spaces; basic concepts and examples.
• Continuous functions.
• Cauchy sequences. Completeness. Completion.
• Convex spaces.
• Compact spaces.
• Product spaces. Tychonoff Theorem.

II - Elements of Analysis:

• Normed spaces. Finite-dimensional spaces. Banach spaces.
• Bounded linear operators. Dual space.
• Spaces with an inner product. Hilbert spaces. Orthonormal bases. Separable Hilbert spaces. Riesz Representation Theorem.

III - Applications:

• Baire Theorem; Abundance of non-differentiable continuous functions; Uniform boundedness principle.
• Banach fixed oint theorem; Existence of solutions for differential equations; Inverse function theorem.
• Arzelà-Ascoli theorem; Spaces of Lipschitz and Hölder functions; Integral operators.
• Stone-Weierstrass theorem; Separability of the space of continuous functions. L^2 space; Fourier series.
• Brouwer fixed point theorem

Mandatory literature

James R. Munkres; Topology. ISBN: 0-13-925495-1
Elon Lages Lima; Espaços métricos. ISBN: 978-85-244-0158-9
Chaim Samuel Honig; Aplicações da topologia à análise
Gueorgui V. Smirnov; Curso de análise linear. ISBN: 972-592-153-4

Teaching methods and learning activities

Exposition of topics in theoretical-practical classes with resolution of illustrative exercises.

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	56,00
Total:	162,00

Eligibility for exams

There will be no attendance control.

Calculation formula of final grade

During the systreet, two written tests will be carried out, each with a rating of 10 (out of 20). The first test takes place in the middle of the semester and the second test takes place at the end of the semester.


The final grade will be the sum, rounded to units, of the grades obtained in the exam.

The exam during the appeal period is made up of two parts corresponding to the tests carried out in the regular period.

Special assessment (TE, DA, ...)

See the course unit Elements of Topology and Analysis, M3027, of 2022/23.