March 24, 2006

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Fixed Point Theory in Fréchet Space
D. P. Dwiggins Systems Support Office of
Admissions
Department of Mathematical Sciences
Analysis Seminar
March 24, 2006
March 24, 2006
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• X is a topological vector space

• S is a closed and convex subset of X

• P S -gt S is a continuous self-mapping

Also, in most settings,

• X is complete

(If not, X may be considered as a dense subset of
its completion.)

• Either X, S, or F is compact

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Fixed Points
The search for fixed points is motivated by

• Finding zeroes of a polynomial

x0 is a zero of p(x) iff x0 is a fixed point of
F(x) x p(x) .

• Finding the null space for an operator

Ax0 0 iff x0 is a fixed point of F(x) x Ax
.

• Finding eigenvectors for an operator

x0 is an eigenvector for an operator A with
corresponding eigenvalue ? ? 0 iff x0 is a fixed
point of F(x) ?Ax , where ? ?-1.
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A TVS is a vector (linear) space endowed with a
topology, under which the operations of vector
addition and scalar multiplication are continuous.
The topology might be given by a norm, a
quasi-norm, or a separable family of semi-norms.
(The chosen topology might also make the TVS
locally convex.)
The topology might be defined in terms of
measure, and the space might be metrizable.
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Assumptions on S

• closed under the topology on X

• convex (a vector property, independent of the
  topology on X)

If X is not assumed to be complete then S must be
(which will be true if S is compact).
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Assumptions on F
If S is not assumed to be compact then F must be
completely continuous i.e. F is both continuous
and compact.
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Finite Dimensional X Brouwers Theorem
Let Sn denote the closed unit ball in Euclidean
space Rn (note Sn is compact). Then any
continuous F Sn ? Sn has a fixed point in
Sn. L. E. J. Brouwer, Math. Annalen 71
(1911) There are many ways to prove this result,
including a purely combinatoric argument using
mappings on finite-dimensional simplices.
Moreover, this theorem also holds if Sn is
replaced by any finite-dimensional Hn which is
homeomorphic to the closed unit ball.
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Infinite Dimensional X Schauder-Tychonov Theory
To extend from finite to infinite dimensional
space, what needs to be determined are the types
of space X for which every continuous
self-mapping F S ? S on any closed convex
compact subset S ? X has a fixed point in S.
Such spaces X are called fixed point spaces, and
Banach spaces (complete normed linear spaces) are
all fixed point spaces. However, the earliest
results were set in spaces which were more
general, and which include Banach space as one
specific example.
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Infinite Dimensional X Schauder-Tychonov Theory
Schauders Theorem (Studia Mathemtaica v.2,
1930) Any complete quasi-normed space is a
fixed point space.
Most authors who cite this theorem assume X to be
a Banach space, which Schauder did not do, and
which he mentioned in a footnote, that the metric
he was using (a quasi-norm) does not possess the
homogeneity of a norm, and thus he was not
working in a B-space.
Tychonovs Theorem (Math. Annalen v.111, 1935)
Any complete locally convex TVS is a fixed point
space.
Since our basic setting assumes a complete TVS,
this theorem might be viewed as best possible.
However, there are quasi-normed spaces which are
not locally convex, and so Schauders Theorem
remains independent.
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Quasi-normed Space versus LCTVS
Let X be a complete metric linear space, for
which the linear operations are continuous with
respect to the metric (i.e. X is a TVS). Then X
becomes a quasi-normed space if the metric is
translation invariant
X would become a Banach space if the metric were
also homogeneous
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Quasi-normed Space versus LCTVS
A LCTVS is a TVS whose topology can be generated
from a separable family of seminorms (Yosida F.
A. pp 23-26). If a LCTVS is metrizable, then its
topology can be obtained from a countable family
of seminorms ?n, from which a quasi-norm
is obtained. However, there are examples of
non-metrizable LCTVS (Yosdia pg 28).
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Quasi-normed Space versus LCTVS
Thus, Tychonovs theorem holds in spaces for
which Schauders theorem does not hold (any
non-metrizable complete LCTVS). Yosida also gives
an example of a complete quasi-normed space which
is not locally convex, meaning there are spaces
in which Schauders theorem holds but not
Tychonovs theorem. The components of this
example appear on pages 38, 117, and 108 (in that
order) of Yosidas text.
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Quasi-normed Space versus LCTVS
Let Q denote the class of all measurable
functions x 0,1 ? C which are defined a.e. on
0,1 (C is the set of complex numbers). Define
a quasi-norm on Q by
Then Q is complete (Yosida pg 38) but is not
locally convex. To prove this, it is first
argued that the dual of Q (denoted by Q?)
consists only of the zero functional (pg 117).
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Quasi-normed Space versus LCTVS
Next, consider the subspace M consisting of all x
? Q such that x(0) 0. Now let y ? Q be given
with y(0) ? 0 (and so y ? M). Then, as a
consequence of the Hahn-Banach theorem (found in
Yosidas text on pg 108), if Q were locally
convex there would be a continuous linear
function f ? Q? such that f(y) gt 1. This
contradicts Q? consisting only of the zero
functional, and so Q cannot be locally convex.
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Wikipedia In functional analysis and related
areas of mathematics, Fréchet spaces, named after
Maurice Fréchet, are special TVSs. They are
generalizations of Banach spaces, which are
complete with respect to the metric induced by
the norm. Fréchet spaces, in contrast, are
locally convex spaces which are complete with
respect to a translation invariant metric, which
may be generated by a countable family of
semi-norms.
Every Banach space is a Fréchet space, which in
general has a more complicated topological
structure due to lack of a norm, but in which
important results such as the open mapping
theorem and the Banach-Steinhaus theorem still
hold.
Other examples of Fréchet spaces include
infinitely differentiable functions on compact
sets (the seminorms use bounds on the kth
derivative over the compact set) and the space
consisting of sequences of real numbers, with the
kth seminorm being the absolute value of the kth
term in the sequence.
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Yosida (F. A. page 52) defines a Fréchet space to
be a complete quasi-normed space (this type of
space was used in the proof of Schauders
theorem), but notes that Bourbaki defines a
Fréchet space as a complete LCTVS which is
metrizable. Every metrizable LCTVS defines a
quasi-norm, but not every quasi-normed space is
locally convex.
Grothendieck (TVS 1973, page 177) states there
are some metrizable LCTVSs which are not
quasi-normable, but uses a quite different
definition of what is a quasi-norm.
It also appears some authors have used the term
Fréchet space to denote a complete LCTVS,
metrizable or not. This is the type of space
used in Tychonovs fixed point theorem.
Finally, some authors have used the term Fréchet
space to denote a space whose topology may be
defined in terms of sequences, without any
reference to a metric or even a vector space.
See Franklin, Spaces in which Sequences
Suffice, Fund. Math. 57 (1965).
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M complete metric space as a TVS, whether or
not the metric is translation invariant
B Banach Space
Schauder space complete quasi-normed space,
locally convex or not
Tychonov space complete LCTVS, metrizable or not
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One way to remove the self-mapping condition F
S S is instead to require Fk S S, where
F2(x) F(F(x)), F3(x) F(F2 (x)), et cetera,
for some k gt 1. Any fixed point theorem using
iterates of the mapping is said to be of
asymptotic type.
Unfortunately it is often just as difficult to
require Fk S S for some k gt 1 as it is for k
1. Instead, consider a sequence of sets S0 Ì
S1 ? S2, where eventual iterates of F map S1 into
S0, and all iterates of F map S1 into S2.
In this setting, F is not a self-mapping on S0,
but eventually every point in S0 ends up back in
S0 as it travels along an orbit of F, and even
early in the orbit the point is never too far
away from where it started.
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Let S0 Ì S1 ? S2 be convex subsets of a Fréchet
space X, with S0 and S2 both compact and S1 open
relative to S2.
Then F has a fixed point in S0.
W. A. Horn, Trans. AMS 149 (1970). Note Horn
assumed X to be a Banach space in his paper.
However, if one lemma is re-written and the
symbol for the norm is everywhere replaced with a
metric then his proof still holds.
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