Exotic normed linear subspaces
Dave L. Renfro
Klee and others have proved about linear subspaces and/or
convex subspaces of infinite-dimensional Banach spaces.
http://groups.google.com/group/sci.math/msg/bae6260afdb9ccdd
http://groups.google.com/group/sci.math/msg/e537f1b019530d9b
I thought it would be useful to dig up the folder I have these
papers in and survey their results.
TABLE OF CONTENTS
1. PRELIMINARIES
2. EXOTIC LINEAR SUBSPACES
3. EXOTIC CONVEX SETS
4. REFERENCES
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1. PRELIMINARIES
In what follows, many of the results were stated and proved
in more general settings, especially the papers below that
were published in the late 1940's and later. However, in the
interest of simplicity, I'll usually stay within the realm
of Banach spaces X. The only topology on X, and the only
metric on X, that will arise in the statements I give will
be those that are derivable from the norm on X.
The term "linear subspace" will mean a subset of X in the vector
space sense, and the term "hyperplane" will mean a translate
of a linear subspace that has co-dimension 1. Hyperplanes in
normed spaces can be characterized as the inverse images of
singletons under linear functionals. That is, if f:X --> R
is linear and r is a real number, then f^(-1)[{r}] is a
hyperplane, and every hyperplane in X arises in this
way for some linear functional f and real number r. If X is
a finite-dimensional normed space, then every linear functional
is continuous, and hence for finite-dimensional normed (= Banach,
in this case) spaces, every hyperplane is closed. In fact,
every proper linear subspace in a finite-dimensional normed
space is both closed and, in a rather strong way, nowhere dense.
Moreover, every proper closed linear subspace in a normed
space is nowhere dense (in a rather strong way).
However, for each infinite-dimensional normed space (neither
completeness nor separability is needed for this), it is known
that discontinuous linear functionals exist. While this doesn't
immediately imply that there exist non-closed hyperplanes,
at least if we only consider the situation from a topological
standpoint (a function will be continuous if the inverse image
of each closed set is closed but, in general, not if we only know
that the inverse image of each singleton set is closed, as any
one-to-one discontinuous function shows), it is known that for
each discontinuous linear functional f, every hyperplane arising
from f is dense in X. Thus, every hyperplane in X is either
closed in X or dense in X. The first two web pages below give
a proof of this, and other proofs (not necessarily different
from a mathematical standpoint) can be found in the google-book
search results below.
http://www.math.unl.edu/~s-bbockel1/928/node22.html
http://tinyurl.com/jhrdn
http://books.google.com/books?q=dense-hyperplane
http://books.google.com/books?q=dense-hyperplanes
Another result worth mentioning is that every Borel linear
subspace in a Banach space (not necessarily separable) is
either first category (i.e. meager) in X or equal to X. This
is proved in Goffman/Pedrick [6] (p. 80).
SUMMARY: Let X be a normed space and V be a proper linear
subspace of X.
** V is closed in X or V is dense in X.
** V closed in X ==> V is nowhere dense in X.
** (X Banach) V Borel in X ==> V is meager in X.
I don't know the historical details of any of the above
results, but I would imagine that all of them (except
possibly the last one) have been known since the
late 1920's.
QUESTIONS: Each of these may be known, easy, both known and
easy, or neither known nor easy. I haven't put
forth much effort towards trying to answer either
of these questions.
** Can (must?) a proper Borel linear subspace of some
(all?) Banach space(s) be meager but not sigma-porous?
NOTE: Proper closed linear subspaces of normed spaces
are porous in a rather strong way. To show nowhere dense,
simply note that if V contains an open ball of center
x and radius r, then V also contains open balls of
center x and radius kx for every k > 0. To show the
stronger property of being porous, make use of F. Riesz's
lemma involving proper closed linear subspaces,
e.g. Goffman/Pedrick [6] (p. 82).
** Can a proper non-Borel linear subspace of some (all?)
Banach space(s) be meager, or even sigma-porous?
NOTE: There exist proper non-Borel linear subspaces
that are not meager, e.g. Hausdorff [7].
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2. EXOTIC LINEAR SUBSPACES
SUMMARY: Let X be a separable Banach space and V be
a proper linear subspace of X.
** V is G_delta ==> V is closed.
** For each countable ordinal alpha >= 1, V can belong
to the additive Borel class alpha and not to the
multiplicative Borel class alpha (and hence, not
to any lower Borel class).
** For each countable ordinal alpha >= 2, V can belong
to the multiplicative Borel class alpha and not to
the additive Borel class alpha (and hence, not to
any lower Borel class).
** For each countable ordinal alpha >= 2, V can belong
to the ambiguous Borel class alpha and not to any
lower Borel class.
** For each positive integer n, V can belong to
the n'th projective class and not to any lower
projective class.
** It is possible for V to not belong to any finite
projective class.
3 September 1931 -- Hausdorff [7] proved that in
every infinite-dimensional Banach space, there
exists a linear subspace that isn't G_delta.
Hausdorff's example was also not first category,
and so by the result I cited from Goffman/Pedrick's
text above, Hausdorff's example was actually not
a Borel set. However, I don't know if Hausdorff
says this in his paper, or even if he was aware of
the result I cited from Goffman/Pedrick's text.
Pettis [14] uses Hausdorff's construction, which
relied on a Hamel basis, to construct pathological
subgroups of R^n.
28 April 1933 -- Mazur/Sternbach [13] proved that in
any infinite-dimensional Banach space, each G_delta
linear subspace is closed and there exist F_sigma_delta
linear subspaces that aren't F_sigma. They also asked
if every infinite-dimensional Banach space has linear
subspaces of arbitrarily high Borel order. According
to Klee [10] (p. 189, footnote 2), Banach announced in
1940 that this is true (Banach died in 1945) and Mazur
presented a proof in a 1957 conference in Zakopane.
A proof is given in Klee [10].
21 June 1933 -- Banach/Mazur [3], improving on one
of the results in Mazur/Sternbach [13], showed that in
each infinite-dimensional Banach space, there exist
F_sigma_delta linear subspaces that aren't G_delta_sigma.
22 June 1933 -- Banach/Kuratowski [2] proved that,
in any separable infinite-dimensional Banach space,
there exist linear subspaces that are CA, CPCA, CPCPCA,
etc. but not A, PCA, PCPCA, etc., respectively (these are
the projective set classifications where A is analytic,
C is complement, and P is projection), as well the
existence of a linear subspace that does not belong
to any of these projective classes. Their proof involves
showing that there exists a co-analytic linear subspace
in C[0,1] (sup norm) having certain properties that is not
Borel, observing that the method works the same way for
the higher level "co-" projective classes, and then making
use of the fact (proved by Banach and Mazur at about
the same time) that every separable Banach space can be
isometrically embedded into C[0,1]. As they were not
able to resolve the problem of whether every (or even
some, although I suspect universal embedding results
would make "some" equivalent to "all") infinite-dimensional
Banach space contains an analytic linear subspace that
isn't Borel (or any of the corresponding higher level
projective versions), they leave this as an open
question. Klee answered this in the affirmative,
with a proof, in [10].
8 December 1958 -- Klee [10] proved that in any
separable infinite-dimensional Banach space we have:
(1) For each countable ordinal alpha >= 1, there exist
dense linear subspaces that belong to the additive
Borel class alpha and not to the multiplicative Borel
class alpha. (2) For each positive integer n, there
exist dense linear subspaces that belong to the n'th
projective class and not to any lower projective class.
26 February 1979 -- Mauldin [12] proved that in any
separable infinite-dimensional Banach space we have:
(1) For each countable ordinal alpha >= 1, there exist
dense linear subspaces that belong to the additive
Borel class alpha and not to the multiplicative Borel
class alpha. (2) For each countable ordinal alpha >= 2,
there exist dense linear subspaces that belong to the
multiplicative Borel class alpha and not to the additive
Borel class alpha. (3) For each countable ordinal
alpha >= 2, there exist dense linear subspaces that
belong to the ambiguous Borel class alpha (i.e. belongs
to both the additive Borel class alpha and to the
multiplicative Borel class alpha) and not to any
lower Borel class.
See Klee [11] for an interesting construction of
a continuum-length chain of exotic linear subspaces
in the Hilbert space L^2[-1,1]. For more recent
miscellaneous results, see Ding/Gao [4] and
Farah/Solecki [5].
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3. EXOTIC CONVEX SETS
SUMMARY: Let X be a normed space.
** For each nonzero cardinal number b <= card(X),
X can be written as a pairwise disjoint union
of b many convex sets, each of which is dense
in X.
** In the previous statement, we can strengthen
"dense in X" to "linearly dense in X". (See
Klee [9] below.)
10 December 1940 -- Turkey [15] (Section 5, p. 101)
proved that each infinite dimensional normed space X
is a disjoint union of two convex sets, each of which
is dense in X.
12 October 1948 -- Klee [8]. Let X be an infinite-dimensional
Banach space and let b be any nonzero cardinal number less
than or equal to card(X). Then X is a pairwise disjoint
union of b many convex sets, each of which is dense in X.
early 1950? -- Klee [9] improved on the previous result
by strengthening "dense" to "ubiquitous". Klee's term
"E is ubiquitous in X" means that the linear closure
of E is equal to X, where the linear closure of a set E
is defined to be E union {x in X: there exists e in E
such that the half-open segment [e,x) is a subset of E}.
[The linear closure of a convex set in R^n is equal
to the closure of that convex set.] Incidentally, Klee's
result makes sense, and holds, in any infinite-dimensional
vector space of the real numbers, and it was in this setting
that Klee proved it. Klee mentions at the end of his paper
that the sets he used to prove the results in Klee [8]
are *not* ubiquitous.
14 July 1951 -- Pettis [14] observed (p. 614, Theorem 4)
that each infinite-dimensional Banach space X is a pairwise
disjoint union of continuum many hyperplanes, each of
which is dense in X. Pettis obtained this result by
considering the sets f^(-1)[{r}], as r varies over the
real numbers, for a discontinuous linear functional f.
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4. REFERENCES
[1] Stefan Banach, "Théorie des Opérations Linéaires",
Monografie Matematyczne #1, 1932, viii + 252 pages.
[Zbl 5.20901; JFM 58.0420.01]
http://www.emis.de/cgi-bin/Zarchive?an=0005.20901
http://www.emis.de/cgi-bin/JFM-item?58.0420.01
http://books.google.com/books?vid=ISBN0828401101
Search in this book = "Sternbach", choose p. 235,
see bottom half of p. 235
http://matwbn.icm.edu.pl/kstresc.php?tom=1&wyd=10
[At the present time, p. 235 isn't among the pages
digitized at this site.]
[2] Stefan Banach and Kazimierz [Casimir] Kuratowski, "Sur la
structure des ensembles linéaires", Studia Mathematica
4 (1933), 95-99. [Zbl 8.31504; JFM 59.0889.02]
Received by the editors on 22 June 1933.
http://www.emis.de/cgi-bin/Zarchive?an=0008.31504
http://www.emis.de/cgi-bin/JFM-item?59.0889.02
http://matwbn.icm.edu.pl/tresc.php?wyd=2&tom=4
[3] Stefan Banach and Stanislaw Mazur, "Eine Bemerkung über die
Konvergenzmengen von Folgen linearer Operationen", Studia
Mathematica 4 (1933), 90-94. [Zbl 8.31602; JFM 59.1073.04]
Received by the editors on 21 June 1933.
http://www.emis.de/cgi-bin/Zarchive?an=0008.31602
http://www.emis.de/cgi-bin/JFM-item?59.1073.04
http://matwbn.icm.edu.pl/tresc.php?wyd=2&tom=4
[4] Longyun Ding and Su Gao, "On separable Banach
subspaces", preprint, 28 April 2006, 6 pages.
http://www.math.unt.edu/~sgao/pub/paper27.pdf
[5] Ilijas Farah and Slawomir Solecki, "Borel subgroups
of Polish groups", Advances in Mathematics 199 #2
(30 January 2006), 499-541. [MR 2006h:03041]
Received by the editors on 30 April 2003.
See p. 503 for some brief historical notes,
or see p. 5 (1'st URL) or p. 6 (2'nd URL):
http://www.math.uiuc.edu/~ssolecki/papers/borpolulm24.pdf
http://www.math.yorku.ca/~ifarah/Ftp/borpolulm-final.pdf
[6] Casper Goffman and George Pedrick, "First Course in
Functional Analysis", 2'nd edition, Chelsea Publishing
Company, 1983, 284 pages. [MR 32 #1540; Zbl 502.46001]
http://www.emis.de/cgi-bin/MATH-item?0502.46001
http://books.google.com/books?vid=ISBN0828403198
Search in this book = "Blumberg", choose p. 80
[7] Felix Hausdorff, "Zur Theorie der linearen metrischen
Räume", Journal für die Reine und Angewandte Mathematik
167 (1932), 294-311. [Zbl 3.33104; JFM 58.1113.05]
Received by the editors on 3 September 1931.
http://www.emis.de/cgi-bin/Zarchive?an=0003.33104
http://www.emis.de/cgi-bin/JFM-item?58.1113.05
http://dz-srv1.sub.uni-goettingen.de/cache/toc/D260807.html
[8] Victor L. Klee, "Dense convex sets", Duke Mathematical
Journal 16 #2 (June 1949), 351-354.
[MR 11,114e; Zbl 41.23303]
Received by the editors on 12 October 1948.
http://www.emis.de/cgi-bin/Zarchive?an=0041.23303
[9] Victor L. Klee, "Decomposition of an infinite-dimensional
linear system into ubiquitous convex sets", American
Mathematical Monthly 57 #8 (October 1950), 540-541.
[MR 12,486b; Zbl 40.06501]
http://www.emis.de/cgi-bin/Zarchive?an=0040.06501
[10] Victor L. Klee, "On the Borelian and projective types of
linear subspaces", Mathematica Scandinavica 6 (1958), 189-199.
[MR 21 #3752; Zbl 88.08502]
Received by the editors on 8 December 1958.
http://www.emis.de/cgi-bin/Zarchive?an=0088.08502
http://www.mscand.dk/issue.php?year=1958&volume=6
[11] Victor L. Klee, "Solution to Monthly Problem #5717", American
Mathematical Monthly 78 #3 (March 1971), 308-309.
Proposed by W. H. Ruckle: "Without using the axiom of choice
(i.e. no Hamel basis) construct a continuum {X_r: 0 < r <= 1}
of linear subspaces of a Hilbert space H which has the following
properties: (a) X_r is dense in H for each r; (b) if r < s,
X_r [is a proper subset of] X_s and X_s/X_r has uncountable
dimension; (c) [the union of all the X_r's is not equal to] H."
Klee's solution: "Represent Hilbert space as L^2[-1,1]. For
each r in [0,1], let X_r consist of all functions f in L^2[-1,1]
such that for some a < r, the restriction of f to [a,1] is
equivalent to a function which assumes only finitely many
values. The set {X_r} fulfills all conditions of the problem."
[12] R. Daniel Mauldin, "On the Borel subspaces of algebraic
structures", Indiana University Mathematics Journal
29 #2 (1980), 261-265. [MR 81i:54027; Zbl 433.22001]
Received by the editors on 26 February 1979.
http://www.emis.de/cgi-bin/MATH-item?0433.22001
[13] Stanislaw Mazur and Leonard Paul Sternbach, "Über die Borelschen
Typen von linearen Mengen", Studia Mathematica 4 (1933), 48-53.
Received by the editors on 28 April 1933.
[Zbl 8.31503; JFM 59.0889.03]
http://www.emis.de/cgi-bin/Zarchive?an=0008.31503
http://www.emis.de/cgi-bin/JFM-item?59.0889.03
http://matwbn.icm.edu.pl/tresc.php?wyd=2&tom=4
[14] Billy James Pettis, "On a vector space construction by
Hausdorff", Proceedings of the American Mathematical Society
8 (1957), 611-616. [MR 19,429d; Zbl 79.32201]
Received by the editors on 14 July 1951.
http://www.emis.de/cgi-bin/Zarchive?an=0079.32201
[15] John Wilder Turkey, "Some notes on the separation of convex
sets", Portugaliae Mathematica 3 (1942), 95-102.
[MR 4,13b; Zbl 28.23202; JFM 68.0238.01]
Received by the editors on 10 December 1940.
http://www.emis.de/cgi-bin/Zarchive?an=0028.23202
http://www.emis.de/cgi-bin/JFM-item?68.0238.01
Dave L. Renfro