Lenguaje Matemático, Conjuntos y Números

Tema Apuntes Problemas Capítulos ref.
1 Lógica Problemas Ch1 [1], Ch1,2,5,6,7 [3]
2 Conjuntos Problemas Ch2 [1], Ch3 [2], Ch3 [3]
3 Relaciones (I), (II), Aplicaciones Ch3 [1], Ch3 [2], Ch8,9 [3]
4 Estructuras algebraicas Ch4 [1]
5 \(\mathbb{N}\), \(\mathbb{Z}\) Problemas Ch5 [1], Ch2,4 [2], Ch4 [3]
6 \(\mathbb{Q}\), \(\mathbb{R}\) Ch6 [1], Ch4 [2]
7 \(\mathbb{C}\) Ch7 [1]

Bibliografía

  1. Delgado Pineda, M. y Muñoz Bouzo, M.J. Lenguaje matemático, conjuntos y números. 2nd Edition, Sanz y Torres.

  2. Tao, T. Analysis I. 4th Edition, Springer Singapore. DOI.

  3. Cummings, J. Proofs: A Long-Form Mathematics Textbook. Independently published. DOI.

Real Analysis I

Unit Notes Problems Chapter ref.
1 \(\mathbb{R}\) Problems
2 Sequences Problems
3 Series Problems
4 Topology in \(\mathbb{R}\) Problems
5 Limits and continuity Problems
6 Differentiability Problems

Bibliography

  1. Abbott, S. Understanding Analysis. 2nd Edition, Springer New York, NY. DOI.

  2. Tao, T. Analysis I. 4th Edition, Springer Singapore. DOI.

  3. Cummings, J. Real Analysis: A Long-Form Mathematics Textbook. 2nd Edition, Independently published. DOI.

  4. Ross, K. A. Elementary Analysis. The Theory of Calculus. 2nd Edition, Springer New York, NY. DOI.

Real Analysis II

Unit Notes Problems Chapter ref.
1 Riemann integral Problems
2 Metric Spaces Problems
2 Continuity in Metric Spaces Problems
3 Sequences of functions Problems
4 Uniform convergence Problems
5 Series of functions Problems
6 Power series Problems

Bibliography

  1. Abbott, S. Understanding Analysis. 2nd Edition, Springer New York, NY. DOI.

  2. Tao, T. Analysis I. 4th Edition, Springer Singapore. DOI.

  3. Tao, T. Analysis II. 4th Edition, Springer Singapore. DOI.

  4. Cummings, J. Real Analysis: A Long-Form Mathematics Textbook. 2nd Edition, Independently published. DOI.

  5. Apostol, T. Mathematical Analysis. 2nd Edition, Pearson.

Multivariate Real Analysis (I, II)

Unit Notes Problems Chapter ref.
1 Calculus in Several Variables Problems
2 Lebesgue measure Problems
2 Lebesgue integral Problems

Bibliography

  1. Tao, T. Analysis II. 4th Edition, Springer Singapore. DOI.

  2. Apostol, T. Mathematical Analysis (2nd ed.). Pearson.

Functional Analysis

Unit Notes Problems Chapter ref.
1 Banach spaces Problems
2 \(C[0,1]\) Problems
3 Hilbert spaces Problems
4 Linear functionals Problems
5 Compact operators Problems

Measure Theory

Unit Notes Problems Chapter ref.
1 Abstract measure space Problems
2 Integration on a measure space Problems
3 Product measures Problems
4 Absolute continuity of measures Problems

Bibliography

  1. Tao, T. Analysis II. 4th Edition, Springer Singapore. DOI.

  2. Stein, E. M., & Shakarchi, R. (2005). Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press. DOI

  3. Royden, A. (1983) Real Analysis. 4th Edition, MacMillan, New York.