第一段原文
8.4. FAMILIES OF FUNCTIONS
The following theorem gives us a characterization for local uniform boundedness when the metric space is a subset ofor.
Theorem 8.4.12 A family of complex valued functionson a subsetofis locally uniformly bounded inif and only ifis uniformly bounded on every compact subset of .
Proof. () This is an immediate consequence of the observation that the closure of a neighborhood inoris compact.
() Supposeis locally uniformly bounded on a domainandandis a compact subset of. Then, for eachthere exists a neighborhood of , and a positive real number, , such that
，
Sincecovers, we know that there exists a finite subcover, say . Then, for,,for all , and we conclude thatis uniformly bounded on.
Remark 8.4.13 Note that Theorem 8.4.12 made specific use of the Heine-Borel Theorem; i.e., the fact that we were in a space where compactness is equivalent to being closed and bounded.
Remark 8.4.14 In our text, an example is given to illustrate that a uniformly bounded
sequence of real-valued continuous functions on a compact metric space need not yield a subsequence that converges (even) pointwise on the metric space. Because
the veriiquest;cation of the claim appeals to a theorem given in Chapter 11 of the text, at
this point we accept the example as a reminder to be cautious.
Remark 8.4.15 Again by way of example, the author of our text illustrates that it
is not the case that every convergent sequence of functions contains a uniformly
convergent subsequence. We offer it as our next excursion, providing space for you
to justify the claims.
Excursion 8.4.16 Let and
.
- Show thatis uniformly bounded in.
- Find the pointwise limit offor.
- Justify that no subsequence ofcan converge uniformly on .
***For (a), observing thatforandfor each yields thatfor. In (b), since the only occurrence ofis in the denominator of each, for each fixed, the corresponding sequence of real goes to 0 as. For (c), in view of the negation of the definition of uniform convergence of a sequence, the behavior of the sequenceat the points allows us to conclude that no subsequences ofwill converge uniformly on.***
Now we know that we donrsquo;t have a “straight” analog for the Bolzano-Weierstrass Theorem when we are in the realm of families of functions in . This poses the challenge of finding an additional property (or set of properties) that will yield such an analog. Towards that end, we introduce define a property that requires “local and global” uniform behavior over a family.
Definition 8.4.17 A familyof complexvalued functions defined on a metric space
is equicontinuous onif and only if
.
Remark 8.4.18 If is equicontinuous on , then eachis clearly uniformly
continuous in.
Excursion 8.4.19 On the other hand, for, show that each function in is uniformly continuous on thoughis not equicontinuous on .
Excursion 8.4.20 Use the Mean-Value Theorem to justify that
is equicontinuous in
The next result is particularly useful when we can designate a denumerable subset of the domains on which our functions are defined. When the domain is an open connected subset of or, then the rationals or points with the real and imaginary parts as rational work very nicely. In each case the denumerable subset is dense in the set under consideration.
Lemma 8.4.21 Ifis a pointwise bounded sequence of complex-valued functions
on a denumerable set, thenhas a subsequencethat converges
pointwise on.
Excursion 8.4.22 Finish the following proof.
Proof. Letbe sequence of complexvalued functions that is pointwise bounded on a denumerable set. Then the setcan be realized as a sequenceof distinct points. This is a natural setting for application of the Cantor diagonalization process that we saw earlier in the proof of the denumerability of the rationals. From the Bolzano–Weierstrass Theorem,bounded implies that there exists a convergent subsequence. The process can be applied to to obtain a subsequence that is convergent.
In general, is such that is convergent and is a
subsequence of each of for. Now consider
*** For, there exists an such that. Then is a
subsequence of from which it follows that is convergent
at. ***
The next result tells us that if we restrict ourselves to domainsthat are compact metric spaces that any uniformly convergent sequence inis also an equicontinuous family.
Theorem 8.4.23 Suppose that is a compact metric space and the sequence
of functions is such that. If converges
uniformly on, then is equicontinuous on.
Proof. Suppose that is a compact metric space, the sequence of functions
converges uniformly onandis given. By Theorem 8.2.3,
is uniforml
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