## Key Elements

MATH 277

#### Formation

BS Computer Sciences

3

6

30

30

0

## Content

#### Objective

The main objective of this course is to complete the base notations of mathematics specially in the domain of the metric topology, in order to show their utility in the different sectors of computer sciences.

#### Content

Chapter 1. « Fundamental properties of IR » Supremum bound of a set – Archimedean property of IR - Absolute value – Density – sequences – Cauchy sequences – IR is complete - Bolzano’s theorem – Topology on IR – Borel Lebesgue theorem. Chapter 2. « Metric spaces » General topology - Distance – Topology of a metric space – Balls – Bounded set – Sequences - Closure of a set – Functions: limits, continuity, uniform continuity and Lipchitz property – Homeomorphism – Product of metric spaces. Chapter 3. « Compact metric spaces » Definition and examples – Properties of continuity function on a compact space –Sequences – limit sup, limit inf - Dini’s theorem - Heine’s theorem. Chapter 4. « Complete spaces » Definition and examples – Convergence problems – fixed theorem – examples of functional complete space – Baire’s lemma. Chapter 5. « Normed vector spaces » Topology of normed vector space – Equivalent norms – Function defined on normed vector space: limits, continuity – Riesz theorem: finite dimension case. Banach space – Space of linear continuous functions – Multilinear continuous function. Chapter 6. « Hilbert spaces » Scalar product – Prehilbertean space – Norms – Cauchy Schwartz inequality – Parallelogram’s formula – Orthogonality – Pythagore’s theorem – Examples of Hilbert space.

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