Abstract
Unless expressly mentioned to the contrary, in this chapter we shall only consider vector functions of areal variable which take their values in a complete normed space over R. When we deal in particular with real-valued functions it will always be understood that these functions are finite unless stated to the contrary.
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Recall (Set Theory, III, p. 144) that a set a of subsets of I is directed with respect to the relation ⊂ if, for any X ∈ F, Y ∈ F, there exists Z ∈ F such that X ⊂ Z and Y ⊂ Z. If S(X) denotes the subset of F formed by the U ∈ F such that U ⊃ X, then the S(X) form a base for a filter on F, called the filter of sections of F the limit (if it exists) of a map f of F into a topological space, with respect to the filter of sections of F, is called the limit off with respect to the directed set F (cf. Gen. Top., I, p. 70 and Gen. Top., IV, p. 348).
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© 2004 Springer-Verlag Berlin Heidelberg
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Theory, E., Spain, P. (2004). Primitives and Integrals. In: Elements of Mathematics Functions of a Real Variable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59315-4_3
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DOI: https://doi.org/10.1007/978-3-642-59315-4_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-63932-6
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