Abstract
Let X be a Hausdorff topological space, and let \({\mathscr {B}}_1(X)\) denote the space of all real Baire-one functions defined on X. Let A be a nonempty subset of X endowed with the topology induced from X, and let \({\mathscr {F}}(A)\) be the set of functions \(A\rightarrow {\mathbb R}\) with a property \({\mathscr {F}}\) making \({\mathscr {F}}(A)\) a linear subspace of \({\mathscr {B}}_1(A)\). We give a sufficient condition for the existence of a linear extension operator \(T_A:{\mathscr {F}}(A)\rightarrow {\mathscr {F}}(X)\), where \({\mathscr {F}}\) means to be piecewise continuous on a sequence of closed and \(G_\delta \) subsets of X and is denoted by \({\mathscr {P}_0}\). We show that \(T_A\) restricted to bounded elements of \({\mathscr {F}}(A)\) endowed with the supremum norm is an isometry. As a consequence of our main theorem, we formulate the conclusion about existence of a linear extension operator for the classes of Baire-one-star and piecewise continuous functions.
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References
J. Borsík, Sums, differences, products and quotients of closed graph functions, Tatra Mt. Math. Publ. 24 (2002), 117–123.
K. Borsuk, Über Isomorphie der Funktionalräume, Bull. Int. Acad. Polon. Sci. (1933), 1–10.
A. Császár, Extensions of discrete and equal Baire functions, Acta Math. Hungar. 56 (1990), 93–99.
A. Császár and M. Laczkovich, Discrete and equal convergence, Studia Sci. Math. Hungar. 10 (1975), 463–472.
J. Doboš, Sums of closed graph functions, Tatra Mt. Math. Publ. 14 (1998), 9–11.
J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367.
R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
R. V. Fuller, Relations among continuous and various non-continuous functions, Pacific Math. J. 25 (1968), 495–509.
F. Hausdorff, Set theory, 2-nd edition, Chelsea Pub. Co., New York, 1962.
T. Husain, Topology and Maps, Plenum Press, New York-London, 1977.
O. F. K. Kalenda and J. Spurný, Extending Baire-one functions on topological spaces, Topology Appl. 149 (2005) 195–216.
B. Kirchheim, Baire one star functions, Real Anal. Exchange 18 (1992/93), 385–399.
K. Kuratowski, Sur les théorèmes topologiques de la théorie des fonctions de variables réelles, C. R. Acad. Sci. Paris 197 (1933), 19–20.
K. Kuratowski, Topology I, Academic Press, New York, 1966.
K. Kuratowski and A. Mostowski, Set Theory, Polish Scientific Publishers, Warszawa, 1976.
R. J. O’Malley, Approximately differentiable functions: The \(r\) topology, Pacific J. Math. 72 (1977), 207–222.
R. J. O’Malley, Baire* 1, Darboux functions, Proc. Am. Math. Soc. 60 (1976), 187–192.
D. E. Peek, Characterizations of Baire* 1 functions in general settings, Proc. Amer. Math. Soc. 95 (1985), 577–580.
H. R. Shatery and J. Zafarani, The equality between Borel and Baire classes, Real Anal. Exch. 30 (2004/2005), 373–384.
A. Wilansky, Functional Analysis, Blaisdell (Ginn), New York, 1964.
M. Wójtowicz and W. Sieg, Affine extensions of functions with a closed graph, Opuscula Math. 35 (2015), 973–978.
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Sieg, W. Linear extensions of some Baire-one functions. Arch. Math. 110, 175–182 (2018). https://doi.org/10.1007/s00013-017-1102-8
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DOI: https://doi.org/10.1007/s00013-017-1102-8