580 Chapter 9 • Sequences and Series of Functions
9.3.6 tell us that a sequence {f n} in C [a , b] converges to some f E C[a, b] iff
it is a Cauchy sequence. Thus, the theory of norm-convergence of functions in
C[a, b] closely parallels the corresponding theory of convergence of sequences of
real numbers.
Definition 9.5.2 A normed vector space Vis said to be complete if every
Cauchy sequence in V converges to an element of V. (From Theorem 2. 7. 7 we
know that, for Archimedean ordered fields, this condition is equivalent to com-
pleteness as we defined it in Chapter l.)
For example, C[a, b] (with the sup norm) is complete, by Theorems 9.2.7
and 9.3.6. Complete normed vector spaces are also called Banach spaces and
are the subject of an entire area of contemporary mathematical research.
There are many normed vector spaces that are not complete. To see this, ob-
serve that in a normed vector space every convergent sequence is also a Cauchy
sequence. By the Weierstrass approximation theorem there are sequences of
polynomials that converge (relative to the sup norm) to a non-polynomial. Thus
there are Cauchy sequences of polynomials in P[a, b] that do not converge to an
element of P[a, b]. Therefore, P[a, b] is not complete. Similar arguments show
that the space of piecewise linear functions on [a, b], and the space of differen-
tiable functions on [a, b] are not complete relative to the sup norm. This is not
the place to discuss completeness relative to the other norms.
We can redo much of Chapters 2, 3, and 5 in the context of a normed vector
space V. For example, we define the c:-neighborhood of a n element v EV to be
the set
N"'(v) = {u EV :llu -vii< c:}.
We say a set A s;;; V is open if V a E A, :3 c: > 0 3 N"'(a) s;;; A , and we say
a set A is closed if A c is open. Similarly, we define an element v E V to be a
cluster point of a set As;;; V if every neighborhood of v contains a member of
A other than v. We define the closure A of a set As;;; V to be the union of A
and the set of all its cluster points, and prove that A is the smallest closed set
containing A. A set A s;;; V is dense in V if every neighborhood of every point
of V contains a member of A. Theorems and proofs concerning these concepts
are virtually the same as the those found in Chapter 3.
Equipped with these preliminaries, we bring our course to an end with a
little razzle dazzle. Don't worry about all the details; just enjoy the ride. As
in Definition 3.4.16, we define a set A s;;; V to be nowhere dense in V if its
closure, A, contains no nonempty open sets. This definition allows us to discuss
first and second category sets, just as we did in Section 5.7 but now in the
context of normed vector spaces. We say that a set A s;;; V is of first category
(or "meager" ) in V if it is the union of a countable collection of nowhere dense
sets; otherwise, we say it is of second category. Of significance here is the
following deep theorem, whose proof we must omit.