First week (week 34)

• Monday: Naive set theory: basic definitions and facts about sets, de Morgan's laws, functions, range and domain of a function, properties of functions: injective, surjective and bijective. See Chapter 1 in the Lecture Notes.

• Tuesday: Left and right inverses, inverses for functions and their characterization in terms of injectivity, surjectivity and bijectivity, countable and uncountable sets, Cantor's diagonal argument. See Chapter 1 in the Lecture Notes.

Second week (week 35)

• Monday: Supremum and infimum of sets, functions and sequences: Definitions, examples and basic statements. See Chapter 1 in the Lecture Notes.

• Tuesday: Definition of real and complex vector spaces, basic examples of vector spaces, subspaces, sum and intersection of vector spaces, direct sums. See Chapter 2 in the Lecture Notes.

Third week (week 36)

• Monday: Norms, basic properties, examples, ||.||_p norms on the space of real n-tuples, Young's inequality, Hoelder inequality See Chapter 3 in the Lecture Notes.

• Tuesday: Minkowski inequality, Triangle inequality for ||.||_p norms of real n-tuples; complex n-tuples, sequences and functions. See Chapter 3 in the Lecture Notes.

Fourth week (week 37)

• Monday: Innerproduct spaces (discussion of the differences between real and complex case), Cauchy-Schwarz, examples, polarization identity.
• Tuesday: Characterization of normed spaces coming from an innerproduct (Theorem of Jordan-von Neumann), orthogonality of vectors, Pythagoras theorem.

Fifth week (week 38)

• Monday: Sequences in normed spaces, bounded subsets in normed spaces. Limit points of a subset of a normed space.
• Tuesday: Cauchy sequence, completeness, least upper bound property of the real numbers.

Sixth week (week 39)

• Monday: Completeness of sequence spaces, pointwise and uniform convergence of continuous functions.
• Tuesday: Completeness of continuous functions on a bounded interval for the supremum norm. Continuity of functions between normed spaces and sequential convergence. Formulation of Banach Fixed Point Theorem.

Seventh week (week 40)

• Monday: Application of Banach's Fixed Point Theorem to Newton's method and to the solution of integral equations, discussion of examples of integral operators (finite-rank operators), Closed, open sets and the boundary of a set: definitions and basic properties.

Eight week (week 41)

• Monday: Characterization of the epsilon-delta definition of continuity in terms of preimages of open sets, bounded operators.
• Tuesday: Continuity of linear maps between normed spaces is equivalent to boundedness, examples (finite-dimensional, matrices between sequence spaces and integral operators), operator norm, completeness of the normed space of bounded mappings for complete target spaces.

Ninth week (week 42)

• Monday: Banach fixed point for solving systems of linear equations, condition number of bounded linear maps, equivalent norms,
• Tuesday: More equivalence of norms. All norms are equivalent in finite-dimensional normed spaces. Examples of closed subspaces and definition of the orthogonal complement.

Tenth week (week 43)

• Monday: Best approximation in Hilbert spaces. The projection theorem. Introduced the projections associated with the projection theorem.
• Tuesday: Riesz' representation theorem. The adjoint operator.

Twelfth week (week 45)

• Monday: Schur form, diagonalizable matrices are dense in the space of all matrices.
• Tuesday: Normal upper triangular matrices are diagonal, basic facts about unitarily diagonalizable matrices.

Eleventh week (week 44)

• Monday: Basic facts about linear transformations between finite-dimensional vector spaces: Change of basis, equivalent matrices, similar matrices, rank-nullity theorem.
• Tuesday: Invariant subspaces, eigenspaces, diagonalizable matrices, existence of eigenvalues for complex matrices.

Thirteenth week (week 46)

• Monday: Spectral theorem for normal operators, positive operators, the matrices AA* and A*A.
• Tuesday: Singular value decomposition, least squares solutions, minimal norm solutions, pseudo-inverse

Fourteenth week (week 47)

• Monday: Generalized eigenvectors and generalized eigenspaces, Jordan blocks, Jordan chains, nilpotent operators, Cayley-Hamilton, minimal polynomial.
• Tuesday: Jordan normal form, exponential of a matrix, solution of vector valued ordinary differential equations, brief discussion of the theorem of Picard-Lindeloef.