Compact Sets in C[a, b]
In class I consider the space C[a, b] of continuous functions on [a, b] equipped with the distance
d(f, g) = max {|f (x) − g(x)|} .
x∈[a, b]
Let E be a subset of C[a, b]. I proved that E is a compact subset of the metric space C[a, b] if both
conditions
1. (“Uniform boundedness”) There is a real number M so that for all f ∈ E,
sup |f (x)| ≤ M.
x∈[a,b]
and
2. (“Equicontinuity”) Given > 0, there is a δ > 0, so that for all x, y ∈ [a, b] and all f ∈ E,
|f (x) − f (y)| < if |x − y| < δ.
are satisfied. The converse is also true and I will prove it in the following. It is know as ArzelaAscoli Theorem.
If a subset E of C[a, b] is uniformly bounded (i.e satisfies 1.) and equicontinuous (i.e. satisfies
2.) then E is compact, which means that any sequence {fj } of elements fn ∈ E has a convergence
subsequence in C[a, b] (meaning a uniformly convergent subsequence on [a, b])
Solution: Let {fj } be a sequence of elements fn ∈ E. Take x1 , x2 , . . . be a dense sequence of
points in [a, b] (e.g. rationals). Now let {f1n } be a subsequence of {fj } such that lim f1n (x1 ) exists
n→∞
(this is possible by Bolzano-Weierstrass theorem for the sequence of real numbers {fj (x1 )} which
bounded by 1.). Now let {f2n } be a subsequence of {f1n } such that lim f2n (x2 ) exists (which is
n→∞
possible by the same argument); repeating this process we have an array of functions fmn such that
each row is a subsequence of the previous row and the mth row converges at x1 , x2 , · · · , xm as n
tends to ∞. Thus
f11 f12 f13 . . .
f21 f22 f23 . . .
f11 f12 f33
..
..
.
.
...
..
.
Taking the diagonal of this infinite matrix we have the subsequence fnn which converges at all
points x1 , x2 . . . (we can choose fmn such that f (x) does not equal fmm (x) for all x on [a, b]).
We now show that the sequence fnn converges uniformly in [a, b]. Indeed, let > 0 be given
and δ > 0 be as in the definition of equicontinuity 2. We choose M large enough such that any
point x ∈ [a, b] has a distance less than δ from at least one of the points x1 , x2 , x3 , · · · xM ; this is
possible since the set of points x1 , x2 , . . . is dense in [a, b] and [a, b] is a compact set so can be
covered by finitely many intervals of length δ. Since fnn converges at each points x1 , x2 , x3 , · · · xM
there exits a Ni such that n, m > Ni
|fnn (xi ) − fmm (xi )| < and take N = max{N1 , N2 , . . . NM } (which is finite since it is the max of finitely many positive
numbers). Note that N depends only on . Thus, using the fact that for x ∈ [a, b] there is an xi ,
i = 1 . . . N such that |x − xi | < δ, by equicontinuity we have for n, m > N
|fnn (x) − fmm (x)| ≤ |fnn (x) − fnn (xi )| + |fnn (xi ) − fmm (xi )| + |fmm (xi ) − fmm (x)| < 3
for all x ∈ [a, b] which means that fnn is a Cauchy sequence in the complete metric space C[a, b]
(or in other words satisfies the uniform Cauchy criterion) and therefore converges to a function
f ∈ C[a, b] (which of course will be in E since fnn ∈ E) and this proves the result.
The Metric Space C 1 [a, b] – Extra Credits Problem
Turn it in by 4pm, Monday, December 14 – use everything you know, my discussion of the space
C[a, b] and the Theorem on uniform convergence and differentiation (I stated it in the last class).
Consider the space C 1 [a, b] of continuously differentiable functions on [a, b] equipped with the
distance
d(f, g) = max |f (x) − g(x)| + |f 0 (x) − g 0 (x)| .
x∈[a, b]
1. Prove that C 1 [a, b] is a complete metric space.
2. Let E be a subset of C 1 [a, b]. Then E is a compact subset of the metric space C 1 [a, b] if and
only if
• (“Uniform boundedness”) There is a real number M so that for all f ∈ E,
sup |f (x)| + |f 0 (x)| ≤ M.
x∈[a,b]
and
• (“Equicontinuity”) Given > 0, there is a δ > 0, so that for all x, y ∈ [a, b] and all
f ∈ E,
|f (x) − f (y)| + |f 0 (x) − f 0 (y)| < if |x − y| < δ.