We prove that the Quasi Differential of Bayoumi of maps between locally bounded F-spaces may not be Fréchet-Differential and vice versa. So a new concept has been discovered with rich applications (see [1–6]). Our F-spaces here are not necessarily locally convex
@article{bwmeta1.element.doi-10_2478_s11533-006-0028-3, author = {Aboubakr Bayoumi}, title = {Bayoumi Quasi-Differential is different from Fr\'echet-Differential}, journal = {Open Mathematics}, volume = {4}, year = {2006}, pages = {585-593}, zbl = {1107.58002}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0028-3} }
Aboubakr Bayoumi. Bayoumi Quasi-Differential is different from Fréchet-Differential. Open Mathematics, Tome 4 (2006) pp. 585-593. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0028-3/
[1] A. Bayoumi: Foundations of Complex Analysis in Non Locally Convex Spaces, Functions theory without convexity condition, North Holland, Mathematical studies, Vol. 193, 2003. | Zbl 1082.46001
[2] A. Bayoumi: “Mean-Value Theorem for complex locally bounded spaces”, Communication in Applied Non-Linear Analysis, Vol. 4(3), (1997).
[3] A. Bayoumi: “Mean-Value Theorem for real locally bounded spaces”, Journal of Natural Geometry, London, Vol. 10, (1996), pp. 157–162. | Zbl 0858.46035
[4] A. Bayoumi: “Fundamental Theorem of Calculus for locally bounded spaces”, Journal of Natural Geometry, London, Vol. 15(1–2), (1999), pp. 101–106. | Zbl 0933.46037
[5] A. Bayoumi: “Mean-Value Theorem for Definite Integral of vector-valued functions of p-Banach spaces”, Algebra, Groups and Geometries, Vol. 22(4), (2005).
[6] A. Bayoumi: “Bolzano’s Intermediate-Value Theorem for Quasi-Holomorphic Maps”, Central European Journal of Mathematics, Vol. 3(1), (2005), pp. 76–82. http://dx.doi.org/10.2478/BF02475656 | Zbl 1069.46508
[7] B.S. Chae: Holomorphy and calculus in normed spaces, Marcel Dekker, 1985. | Zbl 0571.46031
[8] I.J. Corwin and R.H. Szczarba: Multivariable Calculus, Marcel Dekker, 1982.
[9] S. Rolewicz: Metric linear spaces, Monografje Matematyczne, Instytut Matematyczny Polskiej Akademii Nauk, 1972. | Zbl 0226.46001