nets (generalized sequences) in analysis

Skip to first unread message

G. A. Edgar

Apr 17, 2000, 3:00:00 AM4/17/00
I am looking for some references dealing with
nets (generalized sequences) used in analysis (as opposed
to used in general topology.

Gerald A. Edgar

Dave L. Renfro

Apr 18, 2000, 3:00:00 AM4/18/00
G. A. Edgar <>
[sci.math Mon, 17 Apr 2000 13:41:02 -0400]


> I am looking for some references dealing with
> nets (generalized sequences) used in analysis (as opposed
> to used in general topology.
> --
> Gerald A. Edgar

For an undergraduate text, see Beardon's book:

1. Alan F. Beardon, "LIMITS: A New Approach to Real Analysis",
Springer-Verlag, 1997. [QA 300 .B416 1997]
MR 98i:26001 Zbl 892.26003



A distinctive feature of Beardon's book is that it uses
generalized sequences to unify the various limiting ideas
that arise in an undergraduate real analysis course.

I found one review of Beardon's book at

<< Reviewer: from Pittsburgh, PA
November 21, 1997
This book presents the unification of the many versions of
limit processes (sequences, series approaching infinity,
approaching a point, integration, etc.) under the more
general concept of directed sets. The applications comprise
only functions from directed sets into the real line. The
terms 'net' and 'generalized sequence' are never mentioned.
The concept of subnet is not introduced. >>

[The term 'net' appears in the Preface of Beardon's book.]

A slightly longer review of Beardon's book appears on pp. 413-414
of Acta Sci. Math. (Szeged) 65 (No. 1-2), 1999. I have two .gif
scanned versions of this review (1024 x 768 resolution and
800 x 600 resolution) that I can send you if you're interested.

The Zentralblatt (Zbl) review of Beardon's book can be found
by going to


but for your convenience I've reproduced it below. [I don't
have access to on-line Math. Reviews.]

Zentralblatt 892.26003
Beardon, Alan F.
Limits. A new approach to real analysis. (English)
[B] Undergraduate Texts in Mathematics. New York, NY: Springer.
ix, 189 p. DM 58.00; oeS 423.40; sFr. 53.00; \sterling 22.50;
$ 34.95 (1997). [ISBN 0-387-98274-4/pbk; ISSN 0172-6056]

In 1922, E. H. Moore and H. L. Smith presented a theory of
limits seeking to unify and generalize the many limit processes
developed in 19th century. A modification of the Moore-Smith
theory was published by E. J. McShane [Studies in modern
analysis, Math. Assoc. of America (1962)] based on the
concept of directed sets. This concept found its way into the
study of general topology but has not yet been incorporated in
more elementary treatments of real and complex analysis. This
book represents a serious attempt to remedy this situation.
The author's writing style is clear and crisp and (except for
the last chapter) the book is self-contained and well organized.
It includes standard material such as sequences, series,
continuity, differentiation, and Riemann integration. There are
some worked examples and a few exercises.

Like most new books, this one has its share of flaws. Minor
misprints occur on p. 5, line 10 up; on p. 10, line 13 up;
on p. 88, line 6; and on p. 168, line 3 up. The symbols for
upper and lower integrals introduced on p. 153 are confusing
because the upper and lower bars attached to the integral sign
look like minus signs. There are also some logical flaws that
should not have appeared in a book that is otherwise written
with great care. The most serious is on p. 17, where the
author defines a complex number $x+ iy$ as simply another way
of writing $(x, y)$. The reviewer was baffled by this
definition, especially after the careful treatment of ordered
pairs in Chapter 1.\par In Chapter 11, no motivation is
provided for introducing Euler's constant $\gamma$, and no
justification is given for the assertion in Theorem 11.3.1
that $\gamma = 0.5772...$ . Finally, the treatment of the
series and integral for $\pi$ in Section 11.5 makes use of
material outside the scope of the text. This reviewer would
have preferred to see a proof that $\pi$ is the area of a
unit circular disk (a companion to Theorem 11.2.1).

[ Tom M.Apostol (Pasadena) ]

Paul R. Patten (North Georgia College & State University) is
presently using Beardon's book in his course "Introduction to
Real Analysis I" (Spring 2000) at NGCSU. The following web page
is the link under "Outline: II. Limits" that appears on the
homepage of Patten's course, and it (the web page cited just
below) contains some links to web pages having to do with nets.


The notes found in the link above are somewhat brief and give
an outline of Beardon's treatment. In an earlier course, not
using Beardon's text, Patten has some additional notes on
generalized sequences and their use in analysis.

Math 420/620 Outline (Winter 1998)--On the top frame, select
"6.General Theory of Limits (Handout) January 13 & 14". The
notes will appear in the bottom frame --->>>>


The notes at the web page I just gave appear to be at


but for some reason my web browser doesn't show the displayed
mathematical expressions on this third web page.

Heinz Bauschke (Okanagan University College, Canada) used
Beardon's text in the Winter 1999 quarter. [Presently,
Bauschuke is NOT using Beardon's text, but rather the 2'nd
edition of Serge Lang's book UNDERGRADUATE ANALYSIS.] Bauschke
has compiled rather large errata list for Beardon's book which
I have a copy of, if you're interested. [I also have some
additional errata not on Bauschke's list, but I have not yet
combined these lists into a non-duplicating single list.]

At the graduate level generalized sequences are used
extensively in the excellent, but not very well known, text
by McShane and Botts:

2. Edward James McShane and Truman Arthur Botts, REAL ANALYSIS,
D. Van Nostrand Company, 1959. [QA 300 .M28 (also .M248)]
MR 22 #84; Zbl 87 (pp. 46-47)

The Zbl review for McShane and Botts' book can be found by
going to the Zbl web page,


entering *author* = Botts and *title* = real analysis, then
selecting the Zbl citation number link "087.04602", and
finally clicking on the link "Display scanned Zentralblatt
page with this review".

An excellent survey of many subtle variations of nets and
filters that can be found in the literature is given in
Chapter 7 ("Nets and Convergences") of Schecter's Handbook ...

Academic Press, 1997. [QA 300 .S339 1997]
MR 98b:00009; Zbl pre970.42431

A lot of information about Schecter's book can be found at


Also extremely useful is McShane's expository paper:

4. Edward James McShane, "A theory of limits", pp. 7-29 in
STUDIES IN MODERN ANALYSIS, volume 1 of MAA Studies in
Math. series, 1962.
Zbl 151.04801

Listed below are some additional references that you might want
to consult for the use of generalized limits in analysis. The
Zentralblatt (Zbl) reviews can be found by going to


and the Jahrbuch (JFM) reviews can be found by going to


5. E. H. Moore, "Definition of limit in general integral
analysis", Proc. Nat. Acad. Sci. (USA) 1 (1915), 628-632.
[JFM 45.0426.03]

6. E. H. Moore and H. L. Smith, "A general theory of limits",
Amer. J. Math 44 (1922), 102-121.
[JFM 48.1254.01]

7. A. A. Bennett, "Generalized convergence with binary
relations", Amer. Math. Monthly 32 (1925), 131-134.

8. H. L. Smith, "A general theory of limits", National Math.
Magazine (= Mathematics Magazine) 12 (1937-38), 371-379.

9. E. H. Moore and R. W. Barnard, GENERAL ANALYSIS, Mem. Amer.
Philos. Soc. I, Part 1 (1935) and Part II (1939).
[Zbl 013.11605 and Zbl 020.36601]

10. E. J. McShane, "Partial orderings and Moore-Smith limits",
Amer. Math. Monthly 59 (1952), 1-11. [A Chauvenet Prize paper.]
[MR 13 (p. 829); Zbl 046.16201]

Princeton Univ. Press, Annals of Math. Studies 31, 1953.
[MR 15 (p. 19); Zbl 051.29301]

12. E. J. McShane, "A theory of convergence", Canadian J. Math.
6 (1954), 161-168.
[MR 15 (p. 641); Zbl 055.41305]

Finally, you might want to look at Thomson's "local system"
generalization of limits, although my guess is that his
emphasis and applications are not quite the same as yours.
The following give very thorough treatments along with
extensive references to the literature. [Note: Thomson's
book and Thomson's two survey papers (that gave rise to his
book) each contain enough material not present in the other
that you'd want to consult both of them if you decide to
pursue Thomson's work.]

13. Brian S. Thomson, "Differentiation bases on the real line,
I", Real Analysis Exchange 8 (1982-83), 278-442.
[MR 84i:26008a; Zbl 525.26002]

14. Brian S. Thomson, "Differentiation bases on the real line,
II", Real Analysis Exchange 8 (1982-83), 278-442.
[MR 84i:26008b; Zbl 525.26003]

15. Brian S. Thomson, REAL FUNCTIONS, Lecture Notes in Math.
1170, Springer-Verlag, 1985.
[MR 87f:26001; Zbl 581.26001]

Dave L. Renfro

G. A. Edgar

Apr 18, 2000, 3:00:00 AM4/18/00
Dave, thanks a lot for the references. Some of them I did not know

Most of these deal more with the general theory than with uses
in analysis. The most analysis-oriented seem to be
numbers 1, 2, 5, 11. Of course there is also the text of Kelley,

Does anyone know of other expositions using nets in analysis?

Reply all
Reply to author
0 new messages