First week

• Monday. Introduction, practicalities regarding the course. The real numbers system: supremum of a set, the axiom of completeness, the Archimedean property, the density of \(\mathbb{Q}\) in \(\mathbb{R}\).
• Thursday. The real numbers system: the absolute value; sequences of real numbers; bounded sequences, monotone sequences, convergent sequences and the relationship between them; the \(\limsup\) and \(\liminf\) of a sequence, examples; Cauchy sequences.

Second week

• Monday. The real numbers system: equivalence between Cauchy and convergent sequences: Bolzano-Weierstrass theorem; elements of topology.
• Thursday. Elements of topology: open and closed sets, closure, interior, boundary of a set and their characterizations, dense subsets, isolated and accumulation points.

Third week

• Monday. Vector spaces. Normed spaces.
• Thursday. \(l^p\) spaces, Young's inequality, Hölder's inequality, Minkowski's inequality.

Fourth week

• Monday. Inner product spaces (begin).
• Thursday. We have finished the section on inner product spaces.

Fifth week

• Monday. Bounded operators between normed spaces (section 2.1.3 in the notes).
• Thursday. Examples of bounded operators. Banach and Hilbert spaces, absolute convergence criterion for Banach spaces.

Sixth week

• Monday. The proofs of the completeness of \( (\mathbb{R}^n, ||\cdot||_p) \) and \( (l^p (\mathbb{R}), ||\cdot||_p) \). Pointwise and uniform convergence of functions.
• Thursday. Finalizing the proof of the completeness of \( ( C[a,b], ||\cdot||_\infty)\). The proof of the completeness of the space \( (B (X, Y), ||\cdot||_{\text{op}})\) of bounded operators (assuming that \(Y\) is complete).

Seventh week

• Monday. Elements of topology in a normed space. The formulation of the Banach fixed point theorem.
• Thursday. The proof of Banach fixed point theorem. Its application to the Newton's method of solving algebraic equations. A short formulation of the Picard-Lindelöf theorem.

Eighth week

• Monday. A more precise formulation of the Picard-Lindelöf theorem, its proof and examples.
• Thursday. Hilbert spaces: the definition, examples of Hilbert spaces (and of spaces that are not Hilbert), the orthogonal complement of a subspace and its basic properties, the best approximation theorem, the projection theorem.

Ninth week

• Monday. Reformulation of the projection theorem, projections on Hilbert spaces, consequences of the projection theorem, Riesz' representation theorem about

linear functionals on Hilbert spaces, the adjoint of a bounded linear operator, adjoint of the multiplication operator.

• Thursday. Adjoint of a bounded linear operator, existence and basic properties, examples, unitary operators, normal operators and selfadjoint operators.

Tenth week

• Monday. Orthogonality of the kernel of a linear bounded operator on a Hilbert space and the range of its adjoint. Applications: Density of the the range of T, Fredholm's alternative, exemplified with the shift operators.
• Thursday. Orthonormal bases for Hilbert spaces, Bessel's inequality, Parseval's relation, Riesz-Fischer theorem, generalized Fourier expansions, orthogonal projections onto closed subspaces in terms of Fourier coefficients. Continuous linear mappings between normed spaces are exactly the bounded ones.

Eleventh week

• Monday. Basic properties of linearly independent sets, basis, polynomials are an infinite dimensional vector space, dimension formula for the sum of two vector spaces.
• Thursday. Coefficient mapping, matrix of a linear transformation, change of bases, equivalent and similar matrices, existence of eigenvectors, Schur triangularization theorem

Twelth week

• Monday. Diagonizable matrices, linear indenpendence of eigenvectors associated to different eigenvalues, the set of diagonalizable matrices is dense in the set of all matrices, Cayley-Hamilton, spectral theorem for normal matrices.
• Thursday. Spectral theorem for selfadjoint matrices, postive definite matrices, singular value decomposition, polar decomposition for invertible matrices.

Thirteen week

• Monday. Pseudo-inverse, QR decomposition.
• Thursday. Jordan normal form, nilpotent matrix, generalized eigenspaces, minimal polynomial, geometric and algebraic dimension.

Fourteen week

• Monday. Applications of Jordan normal form, including the solution of ordinary differential equations, Equivalence of norms, and that all norms on a finite-dimensional vector space are equivalent, but not so for infinite-dimensional vector spaces. Matrix norms.
• Thursday. Metric spaces: discrete metric, Hamming distance, seminorms and the construction of metric sapces via a countable family of seminorms (aka Frechet spaces), metric on the space of smooth functions, complete metric spaces.