#### Objective

1. Know that a vector normed space is a metric space.
Recognize some properties of distances and norms.
Recognize the bounded and non bounded sets.
Know the properties of open and closed balls and spheres.
2. Know the properties of topologically and uniformly equivalent distances.
3. Know the metric topology and its concepts. Know the relation between a neighborhood of a point and an open ball centered at this point. Know the notion of interior, adherent, isolated and limit point.
4. Know the properties of open and closed balls in metric spaces and those specific to vector normed spaces.
5. Know the notions of open and closed sets in a metric space. Know the three equivalent norms in a finite product of metric spaces.
6. Know the notion of sequence in a metric space, bounded sequence, convergent sequence, divergent sequence and Cauchy sequence. Know that a convergent sequence possesses a unique limit and that a Cauchy sequence is bounded and that each convergent sequence is a Cauchy sequence.
7. Know the notions of series, convergent series, normally convergent series and semi-convergent series in a vector normed space.
8. Know the notions of limit and continuity of a function from a metric space into another metric space. Know how to prove the continuity of a function through the use of sequences, open or closed sets. Know the notions of uniformly continuous functions, Lipschitz and contracting functions.
9. Know the notions of homeomorphism and isometry and related theorems.
10. Know the notions of linear continuous mappings on a vector normed space and their properties.
11. Know the notion of complete metric space and the fixed point theorem. Know the Banach spaces and the corresponding theorems about series.
12. Know the notion of compact metric space through the use of open coverings and sequences. Know the Borel-Lebesgue theorem, the Bolzano-Weierstrass and theTychonoff theorems concerning the finite product of compact metric spaces. Know that that all norms are equivalent on a vector normed space of finite dimension and that each vector normed space of finite dimension is a Banach space.

#### Content

I. Vector normed space – Metric space.
1. Norm, equivalent norms, properties. Convex set.
2. Distance, distance associated to a norm. Bounded set, balls and spheres.
II. Topology on a metric space (resp on a normed space).
1. Open, closed, neighborhood and topology in a metric space.
2. Interior, adherent, limit, frontier, exterior and isolated point of a subset. Dense subset. Separable space.
3. Theorems and properties.
4. Subspace and product space.
III. Sequences in a metric space
1. Definition of a sequence. Convergent, divergent, bounded sequences subsequences, adherence values.
2. Cauchy sequences and their properties.
3. Series in a normed space : Convergent, divergent and normally convergent series.
IV. Continuous functions.
1. Definition of a limit. Continuity and uniform continuity. Topologically and uniformly equivalent distances. Lipshitz and contracting functions.
2. Continuity and product space.
3. Homeomorphism, isometry and their properties.
4. Linear continuous mappings. Isomorphism and linear isometry.Topological dual. Bilinear mappings.
V. Complete metric space.
1. Completion, complete subspace and product space.
2. Fixed point theorem. Extension by continuity theorem.
3. Banach space: series in a Banach space and the space
£(X,Y) = f: X Y : f linear and continuous
VI. Compact metric space.
1. Definition through sequences and definition through open coverings. Properties and theorems.
2. Lebesgue and Tychonoff theorems.
3. Vector normed space of finite dimension.
4. Locally compact space.
Comment:
This course plays a double role in that it constitutes a generalization of notions already treated in real analysis and a prelude to purely topological notions to be developed in following courses. Thus, students have to get familiarized with the vocabulary of topology. We will stress in the course on that each notion must be followed by illustrative examples picked in real analysis. We will stress also on the major role of exercises in the process of assimilation of the concepts of this course. The definitions of the continuity using the - or the convergent sequences are the same in metric spaces. The mean-value theorem, the maximum value theorem and the uniform continuity theorem are very rich tools for diverse practical applications. We will indicate that notions such that connectivity, arc connectivity or compactness are topological invariants. As for vector normed spaces, we will just enlighten their topological properties taking as a model. The equivalence among all norms on must be clarified. It must be noticed that the norm is a 1-Lipschitz function and that each vector normed space of finite dimension is complete and homeomorphic to .