Lineability, spaceability, and latticeability of subsets of C([0, 1]) and Sobolev spaces

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This work is a contribution to the ongoing search for algebraic structures within a nonlinear setting. Here, we shall focus on the study of lineability of subsets of continuous functions on the one hand and within the setting of Sobolev spaces on the other (which represents a novelty in the area of research).
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1. Aron, R.M., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Lineability and spaceability of sets of functions on R. Proc. Am. Math. Soc. 133(3), 795–803 (2005). 2. Aron, R.M., García-Pacheco, F.J., Pérez-García, D., Seoane-Sepúlveda, J.B.: On dense-lineability of sets of functions on R. Topology 48(2–4), 149–156 (2009). 3. Aron, R.M., Bernal González, L., Pellegrino, D.M., Seoane-Sepúlveda, J.B.: Lineability: The Search for Linearity in Mathematics, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2016) 4. Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am. Math. Soc. (N.S.) 51(1), 71–130 (2014). 5. Bernal-González, L., Fernández-Sánchez, J., Seoane-Sepúlveda, J.B., Trutschnig, W.: tHighly tempering infinite matrices II: from divergence to convergence via Toeplitz-Silverman matrices. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(4), Paper No. 202, 10 (2020). 6. Bonilla, A., Muñoz-Fernández, G.A., Prado-Bassas, J.A., Seoane-Sepúlveda, J.B.: Hausdorff and Box dimensions of continuous functions and lineability. Linear Multilinear Algebra 69(4), 593–606 (2021). 7. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2011) 8. Ciesielski, K.C., Natkaniec, T.: Different notions of Sierpinski–Zygmund functions. Rev. Mat. Complut. 34(1), 151–173 (2021). 9. Ciesielski, K.C., Seoane-Sepúlveda, J.B.: A century of Sierpinski–Zygmund functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(4), 3863–3901 (2019). 10. Ciesielski, K.C., Seoane-Sepúlveda, J.B.: Differentiability versus continuity: restriction and extension theorems and monstrous examples. Bull. Am. Math. Soc. (N.S.) 56(2), 211–260 (2019). 10.1090/bull/1635 11. Ciesielski, K.C., Gámez-Merino, J.L., Mazza, L., Seoane-Sepúlveda, J.B.: Cardinal coefficients related to surjectivity, Darboux, and Sierpi ´nski–Zygmund maps. Proc. Am. Math. Soc. 145(3), 1041–1052 (2017). 12. Conejero, J.A., Fenoy, M., Murillo-Arcila, M., Seoane-Sepúlveda, J.B.: Lineability within probability theory settings. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111(3), 673–684 (2017). 13. de Amo, E., Díaz Carrillo, M., Fernández-Sánchez, J.: Singular functions with applications to fractal dimensions and generalized Takagi functions. Acta Appl. Math. 119, 129–148 (2012). 14. Dovgoshey, O., Martio, O., Ryazanov, V., Vuorinen, M.: The Cantor function. Expo. Math. 24(1), 1–37 (2006). 15. Falcó, J., Grosse-Erdmann, K.-G.: Algebrability of the set of hypercyclic vectors for backward shift operators. Adv. Math. 366, 107082 (2020). 16. Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications, 3rd edn. Wiley, Chichester (2014) 17. Fernández Sánchez, J., Trutschnig, W.: A note on singularity of a recently introduced family of Minkowski’s question-mark functions. C. R. Math. Acad. Sci. Paris 355(9), 956–959 (2017). (English, with English and French summaries) 18. Fernández-Sánchez, J., Rodríguez-Vidanes, D.L., Seoane-Sepúlveda, J.B., Trutschnig, W.: Lineability, differentiable functions and special derivatives, Banach J. Math. Anal. 15(1), Paper No. 18, 22 (2021). 19. García, D., Grecu, B.C., Maestre, M., Seoane-Sepúlveda, J.B.: Infinite dimensional Banach spaces of functions with nonlinear properties. Math. Nachr. 283(5), 712–720 (2010). 20. Jiménez-Rodríguez, P.: c0 is isometrically isomorphic to a subspace of Cantor–Lebesgue functions. J. Math. Anal. Appl. 407(2), 567–570 (2013). 21. Kunze, H., La Torre, D., Mendivil, F., Vrscay, E.R.: Fractal-Based Methods in Analysis. Springer, New York (2012) 22. Oikhberg, T.: A note on latticeability and algebrability. J. Math. Anal. Appl. 434(1), 523–537 (2016). 23. Schwartz, L.: Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX–X. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris (1966) (French) 24. Seoane-Sepúlveda, J.B.: Chaos and lineability of pathological phenomena in analysis, ProQuest LLC, Ann Arbor (2006). Thesis (Ph.D.), Kent State University 25. Shidfar, A., Sabetfakhri, K.: Notes: on the continuity of Van Der Waerden’s function in the holder sense. Am. Math. Mon. 93(5), 375–376 (1986). 26. Trutschnig, W., Fernández Sánchez, J.: Copulas with continuous, strictly increasing singular conditional distribution functions. J. Math. Anal. Appl. 410(2), 1014–1027 (2014).