Abstract
The purpose of this Book is the elementary study of the infinitesimal properties of one real variable; the extension of these properties to functions of several real variables, or, all the more, to functions defined on more general spaces, will be treated only in later Books.
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The elements (or vectors) of a vector space E over a commutative field K will usually be denoted in this chapter by thick minuscules, and scalars by roman minuscules; most often we shall place the scalar t to the right in the product of a vector x by t, writing the product as xt; on occasion we will allow ourselves to use the left notation tx in certain cases where it is more convenient; also, sometimes we shall write the product of the scalar 1/t (t ≠ 0) and the vector x in the form x/t.
We recall that a norm on E is a real function ∥x∥ defined on E, taking finite non-negative values, such that the relation 11x11 = 0 is equivalent to x = 0 and such that>∥x + y∥ ≤ ∥x∥ + ∥y∥ and ∥xt∥ = ∥x∥. |t| for all t ∈ K (t being the absolute value of t in K).
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© 2004 Springer-Verlag Berlin Heidelberg
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Theory, E., Spain, P. (2004). Introduction. In: Elements of Mathematics Functions of a Real Variable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59315-4_1
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DOI: https://doi.org/10.1007/978-3-642-59315-4_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-63932-6
Online ISBN: 978-3-642-59315-4
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