Transcript Slide 1

Fixed Point
Theory in
Fréchet Space
D. P. Dwiggins
Systems Support
Office of Admissions
Department of
Mathematical
Sciences
Analysis Seminar
March 24, 2006
Basic Setting
• X is a topological vector space
• S is a closed and convex subset of X
F :: S ->
 S is a continuous self-mapping
•P
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
Fixed Points
A fixed point of F : S  S is any point
x0  S such that F ( x0 )  x0 .
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.
Topological
Vector Space
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 (The chosen topology
might also make the
a norm, a quasi-norm, or a
TVS locally convex.)
separable family of semi-norms.
The topology might be defined in terms of
measure, and the space might be metrizable.
Assumptions on S
S  X is always assumed to be:
• closed under the topology on X
• convex (a vector property, independent
of the topology on X):
 x, y  S t 0,1 the convex
combination tx  (1  t )y is in S.
If X is not assumed to be complete then S
must be (which will be true if S is compact).
Assumptions on F
F is continuous and F ( S )  S . (self-mapping condition)
(This is the usual assumption, with other possible
assumptions of the type S  F (S ) or F k (S )  S
for some iterate Fk – this defines asymptotic FPT.)
If S is not assumed to be compact then
F must be completely continuous; i.e.
F is both continuous and compact.
F is compact if F(A) is compact for every bounded A  S .
As with the self-mapping condition, this assumption
might also be replaced with some alternative, such as
requiring Fk to be compact for some k > 1.
Finite Dimensional X
Brouwer’s 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 finitedimensional simplices. Moreover, this theorem also holds
if Sn is replaced by any finite-dimensional Hn which is
homeomorphic to the closed unit ball.
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.
Infinite Dimensional X
Schauder-Tychonov Theory
Schauder’s 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”.
Tychonov’s 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 Schauder’s Theorem remains independent.
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, y)   ( x  y,0) x, y  X
X would become a Banach space if the
metric were also homogeneous:
 ( x,0)   ( x,0) x  X   0
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

n ( x, 0)
1
xq  n 
1  n ( x, 0)
n 1 2
is obtained. However, there are examples
of non-metrizable LCTVS (Yosdia pg 28).
Quasi-normed Space
versus LCTVS
Thus, Tychonov’s theorem holds in spaces for
which Schauder’s 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
Schauder’s theorem holds but not Tychonov’s
theorem. The components of this example
appear on pages 38, 117, and 108 (in that
order) of Yosida’s text.
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
This also gives a
translation invariant
1
x(t )
metric  by defining
xq 
dt
 ( x, y )  x  y
0 1  x(t )
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).
q
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 Yosida’s text on pg 108), if
Q were locally convex there would be a
continuous linear function f  Q such that
f(y) > 1. This contradicts Q consisting only
of the zero functional, and so Q cannot be
locally convex.
What is a
Fréchet Space?
Wikipedia: In functional analysis and related areas of mathematics,
Fréchet spaces, named after Maurice Fréchet, are special TVS’s. 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 BanachSteinhaus 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.
What is a
Fréchet Space?
Yosida (F. A. page 52) defines a Fréchet space to be a complete quasinormed space (this type of space was used in the proof of Schauder’s
theorem), but notes that “Bourbaki” defines a Fréchet space as a complete
LCTVS which is metrizable. Every metrizable LCTVS defines a quasinorm, but not every quasi-normed space is locally convex.
Grothendieck (TVS 1973, page 177) states there are some metrizable
LCTVS’s 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
Tychonov’s 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).
What is a
Fréchet Space?
S
T
M
M = complete metric
space as a TVS, whether
or not the metric is
translation invariant
B
B = Banach Space
Schauder space = complete quasi-normed space, locally convex or not
Tychonov space = complete LCTVS, metrizable or not
Wikipedia:
Dwiggins:
F  S T
F  S T
Asymptotic
Fixed Point Theory
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 > 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 > 1 as it is for k = 1. Instead, consider a sequence
of sets S0  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.
Asymptotic FPT
Horn’s Theorem
Let S0  S1  S2 be convex subsets of a Fréchet space X,
with S0 and S2 both compact and S1 open relative to S2.
Let F : S2  X be a continuous map such that, for some m > 1,
(1) F k (S1 )  S2 , 1  k  m, and
(2) F k (S1 )  S0 , m 1  k  2m.
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.
S1
S0