Undergraduate Analysis pp 160-192 | Cite as

# Limits

Chapter

## Abstract

A number of notions developed in the case of the real numbers will now be generalized to normed vector spaces systematically. Let
we have
.

*S*be a subset of a normed vector space. Let*f*:*S*→*F*be a mapping of S into some normed vector space*F*, whose norm will also be denoted by ||. Let*υ*be adherent to*S*. We say that the**limit of***f*(*x*)**as***x***approaches***υ***exists**, if there exists an element*w*∈*F*having the following property. Given*ε*, there exists*δ*such that for all*x*∈*S*satisfying$$\left| {x - \upsilon } \right| < \delta $$

$$\left| {f(x) - w} \right| < \varepsilon $$

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## Copyright information

© Springer Science+Business Media New York 1997