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Notes for the course Analysis I, MATE–2201 Universidad de los Andes, 2010–20
STEFANO FERRI firstname.lastname@example.org
1. Introduction to the problem
In these notes we discuss ways to deﬁne convergence for a sequence of functions (at ﬁrst werestrict ourselves to functions from R to R) in a meaningful way. Namely, we would like to be able to “approximate” a function by a sequence of functions (i.e. to regard a function as the limit of the sequence) and use the sequence in our abstract arguments much in the same way as we did in the Calculus courses when we wrote (or even deﬁned) the functions sin, cos and exp as the sum of a series.Suppose that (fn )n is a sequence of functions fn : R −→ R, the most naive way to deﬁne a concept of limit for the sequence of functions is to write: (lim fn )(x) := lim fn (x). The problem is that this way of deﬁning limits does not behave well under taking iterated limits. In particular, we have to face the unpleasant fact that with the above deﬁnition of limit a sequence of continuous functions doesnot need to converge to a continuous limit, a sequence of derivable functions does not need to converge to a derivable one and a sequence of integrable functions does not need to converge to a integrable one. Moreover, even when the limit of a sequence of continuous functions is indeed continuous or a limit of a sequence of derivable functions is derivable or a limit of integrable functions isintegrable we cannot freely interchange limits and we cannot derivate or integrate a sequence term by term. Probably, the simplest example which shows this is the following. 1.0.1 Example. Let n ∈ N and deﬁne fn : [0, 1] −→ R by fn (x) := xn . It is clear that for any n ∈ N the function fn is continuous (in fact, even diﬀerentiable) on [0, 1] and that, for x ∈ [0, 1) we have that limn fn (x) = 0. Onthe other hand, limn fn (1) = 1, so the pointwise limit of (fn )n is the function: 0 if 0 ≤ x < 1 and 1 if x = 1, which is not continuous in 1.
Preliminary version: do no distribute!
2 In class we presented another example which also illustrates the problem and which is just sligtly more complicated to write but possibly easier to picture. 1.0.2 Example. Let (fn )n be a sequence offunctions with fn : [0, 1] −→ R deﬁned by 1 if x ∈ [0, 1 − n ) 0 2 1 1 fn := n(x − ) + 1 if x ∈ ( 1 − n , 1 ) (1.) 2 2 n 1 1 if x ∈ ( 2 , 1].
1 It is not diﬃcult to see that when x ∈ [0, 2 ) we have that lim fn (x) = 0 1 and when x ∈ [ 2 , 1] we have that lim fn (x) = 1, so the pointwise limit of this sequence is not continuous. More involuted examples can be constructed which show that thepointwise limit of a sequence of diﬀerentiable (respectively integrable) functions need not be diﬀerentiable (respectively integrable) and, even if it happens to be, the limit of the derivatives (respectively the integrals) is not necessarily the derivative (respectively the integrals) of the limit. Because of these problems we shall deﬁne several diﬀerent forms of convergence. Since we shall usethe framework of normed space we start by giving a short introduction to this subject.
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2. Normed spaces
2.1. First deﬁnitions and examples
We start with the deﬁnition of some basic concepts in the theory of normed spaces. In what follows all vector spaces will be on a ﬁeld F which is either R or C. 2.1.1 Deﬁnition. Let X be a vector spaceover F, a norm on X is a function · : X −→ R
with the following properties:
(a.) for every x ∈ X, X ≥ 0 and we have x = 0 if and only if x is the zero vector; (b.) for every x ∈ X and every α ∈ F we have that αx = |α| · x ; and (c.) for every x, y ∈ X, x + y ≤ x + y . 2.1.2 Remark. If X is a normed space with norm to check that X with the distance deﬁned by d(x, y) := x − y is a metric space...