Xtewey
-^KSS. INST.
JUL
ALFRED
P.
SLOAN SCHOOL OF MANAGEMENT
TOPOLOGICAL SEMIVECTOR SPACES;
*
CONVEXITY AND FIXED POINT THEORY
QouC
Prem Prakash^
602-72
Murat R. Sertel
May, 1972
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
,MBRIDGE, MASSACHUSETTS
24
TE^.
1972
MASS. INST. T£CH.
JUL
22
1972
TOPOLOGICAL SEMIVECTOR SPACES:
CONVEXITY AND FIXED POINT THEORY
Prem Prakash^
602-72
Murat R. Sertel
May, 1972
(Also carries Working Paper #112-72 from the Graduate School
of Management, Northwestern University, Evanston, Illinois.)
We would like to thank Profs. James R. Munkres (M.I.T.) and Alexander
D. Wallace (U. Florida) for short-circuiting some early fallacious
conjectures. We are indebted to Prof. M. P. Schutzenberger (U. Paris)
for a remark which compelled us to seek and formulate the notion of a
"left skew semifield" [Prakash & Sertel, 1972] as the general sort of
object over which to define semivector spaces. To Prof. T. Bowman
(U. Florida) we are grateful for pointing out an error in an earlier
draft of Section
2.
We would like to acknowledge the support provided by Prof. Z. S.
Zannetos (M.I.T.) through a NASA grant NGL22-009-309 for the work of
M.R.S. at the Management Information for Planning and Control Group,
Alfred P. Sloan School of Management, M.I.T.
RECEIVED
JUL 24 1972
I.
T.
LIbKAKiES
TOPOLOGICAL SEMIVECTOR SPACES:
CONVEXITY AND FIXED POINT THEORY
Prem Prakash
0.
Murat R. Sertel
INTRODUCTION
Without speaking too roughly, (topological) semivector spaces
are to (topological)
semigroups as (topological) vector spaces are
to (topological) groups.
Recalling J. L. Kelley's [l955,
p.
lio]
remark indicating the importance of convexity arguments as the
basis of results distinguishing the theory of topological vector
spaces from that of topological groups, one may expect to see
convexity playing
corresponding role more generally in
a
distinguishing the theory of topological semivector spaces from
that of topological semigroups.
Meanwhile, the results presented
here may be taken to illustrate that much of the power of convexity
properties is preserved in less stringent contexts than that of
a
vector space structure.
The notion of
a
"semivector space" was first introduced in
[Prakash & Sertel, 1970a]; some structural aspects of such spaces
were examined in [Prakash & Sertel, 1971a].
[Prakash & Sertel,
1972] is devoted entirely to the structure of semivector spaces.
635556
Finally, an application of some of the fixed point theory of the
present paper in generalizing existence results for equilibria in
various types of social systems, systems which include economies
and games, will be found in [Prakash, 1971
],
[Sertel, 1971
]
and
[Prakash & Sertel, 1971b].
In this paper we start by defining topological semivector
spaces.
After briefly considering some problems in "strengthening"
their topology, we extend the notion of convexity to such spaces.
We then identify
a
hierarchy of local convexity axioms for such
spaces and present some simple facts about products of spaces
having various local convexity properties.
Next, we illustrate
how the spaces of concern arise naturally as certain hyperspaces
of topological (semi-) vector spaces.
Finally, we establish a
number of fixed point and minmax theorems for topological
semivector spaces with various local convexity properties.
The
fixed point results which we obtain here will be seen to generalize
central fixed point theorems (for topological vector spaces) due
to S. Kakutani [l94l], H. F. Bohnenblust & S. Karlin [l950] and
K. Fan [l952j, which,
in turn, are generalizations of results due
to L. E. J. Brouwer [l912], J. Schauder [l930] and A. Tychonoff
[1935], respectively.
1.
PRELIMINARIES
Into this section we compress a quick review of some basic
notions introduced and discussed in detail elsewhere [e.g., see
Prakash & Sertel, 1972].
In defining "semivector" spaces, the notion of a "left skew
By a left skew semifield we mean
semlfield" serves as a catalyst.
a
<©,+,.>
bimonoid
distinct from its identity
with identity
<0,
+>
a
<0,
+,
.
>
.
+ y)
(p
<
0,
.
>
(P
+ y)
•
Ct.
is commutative,
semifield .
>
is a comnutative semigroup
<
left action of
=
with zero
is a group
O- •
P +
O-
Y
.
©,
.
>
on
A left skew
•
will be called a skew semifield iff the
< 0,
(unitary) right action, too, of
homomorphic:
.
+>
<@,
1,
and the (unitary)
0,
is homomorphic:
semifield
< 0,
in which
=
^.
+ y.
0.
(X
.
>
<
on
is
f >,
This certainly obtains if
.
in which case we term
Notably, the set
0,
<0,
+,
.
>
a
of non-negative reals under the
R
usual addition and multiplication provides
a
useful example of
semifield, one which we call the usual real semifield
.
a
Whenever
considered as a topological space, the set of non-negative reals
will be understood to carry the Euclidean topology.
1.0
Definition
:
Let
skew semifield, and
be a topological space,
<S,
®>
(not necessarily Hausdorff).
a
<0,
f,
.
>
a
left
commutative topological semigroup
Let
f
:
©
x
s
-»
S
,
where we denote
be a continuous map satisfying
Y( \, s) = X s,
Axiom
1
Axiom
2
Axiom
3
for all
X(|as) = (X.|a)s
s
e
(action)
(unitariness)
s
X(s®t) =X.s®Xt
X,
li
and
e
s,
(homomorphism)
e S.
t
will be called
S
topological semivector space over
,
a
convex iff
contains
the usual real semifield,
1,1
Remark
with
a
:
Clearly, when
(1)
a
is equipped
left skew semifield
topology yielding its operations continuous (for example,
the discrete topology)
,
may be viewed as
topological
a
Hence our comment, at the outset,
semivector space over itself.
anticipating the catalytic role of the notion of left skew
semifields in defining topological semivector spaces.
In view of the fact that
(2)
<0,
.
>
is a group with
zero, it is easily seen that the second axiom of the definition
above is equivalent to the requirement that
obtain for each
e S.
is an action merely of
by defining
should
At this stage it might seem strange that, although
(3)
Y
s
= s
1 s
S
<
0,
.
>
on
S,
we invoke
to be over a left skew semifield.
however, to make full use of the operation
(+)
,
<
0,
+>
We intend,
e.g., in
speaking of "convexity" of sets in topological semivector spaces.
,
The reason for adopting the qualifier 'convex' for
(4)
when
S
contains the usual real semifield will become
apparent immediately after we define convexity in 3.1.
Given
topological semivector space
the restriction
X e 0,
transition of
(X-)
of
a
Y
to
From Axiom
S.
is an endomorphism of
S
Y
of
S,
3
S
{X} x
over
S
0,
will be called the
we see that each transition
and from the continuity of
is immediate that each transition is also continuous.
when
X
?^
¥
0,
for each
is an automorphism of
S,
Y
it
In fact,
and, writing
(J.
=
—
X
where
—
denotes the inverse of
X
under multiplication
(.),
A.
have
^
map.
It
(0
\
continuous, whereby
follows that
l0|) X
S.
Y
Y
is both an open and a closed
is an open map when restricted to
we
2.
STRENCTHENABILITY OF THE TOPOLOGY
If
<S,
©>
with identity
is a commutative Hausdorff topological semigroup
it is possible to strengthen the topology on
e,
without destroying the continuity of
the neighborhood (nbd) system of
U ®
is now open whenever
s
U
e
"
with identity
S
strong " topology on
S
is unaltered, while
is open in
De Miranda, 1964; Theorem 3.2.13],
semi vector space
in such a way that
®,
Given
e,
(s
S
a
S,
(i)
(ii)
[Paalman-
e S)
Hausdorff topological
by a
"
strengthened " or
we will mean one which satisfies
(i)
and
as just stated.
(ii)
Given
a
Hausdorff (topological) semivector space
we may now ask whether--or when--there exists
topology on
space.
S
under which
(Of course,
S
remains
a
a
S
over
0,
strengthened
topological semivector
in a topological vector space the topology is
already a strengthened version of itself).
Having Paalman-De Miranda's result as stated above, the
question clearly boils down to whether the continuity of
be preserved under a strengthened topology on
S.
Y
can
Although we are
unable to assert in general when this can be done and when it
cannot, we recognize a research problem here and offer the following
as an example of where it cannot be done even though the space
whose topology is to be strengthened is, as the reader may check,
a
pointwise convex (see 3.2) topological semivector space with
identity and with
locally convex (see 4.1 and 6.5), etc.
3
2.1
topology which is locally compact, metrizable,
a
Example
let
Let
:
be the real field with the usual topology, and
F
be the topological group of the reals (under the usual
R
addition and with the usual topology)
closed intervals of
Defining
operations is
= e
(a,
b,
(x,
d 6 R;
c,
e
y)
X e F)
,
^ 7
V.
attention to any non-singleton
Y: Fx
restriction
KJliR)
jp
|
-»
P £ ?C,i(R)
of
J^.2.(R)
11.
,
1(
1{
1(
® XP
under
that
© U
be an open nbd of
XP
of
<
for some
contains
Y_
^i
<
for some
U e
P)
,
^(.
Then
Tp
with these
and has
JOJ
by
?('.2(R)
of
Fix
e.
The fact is that Y
K^(R)
•
Now the inverse image of
0.
but it contains no
P =
diam(— P © —
impossible, since
This shows that
(X,
>
X
and
is
For
and consider the (basic) open nbd
e,
<
(For suppose that
X.
F
d]|
[c,
and consider the
not continuous under the strengthened topology on
let
x
to be (basic) open for each
and for each "originally" open nbd
?C.2.(R)
b]
[a,
Now strengthen the topology on
as its identity.
declaring the translates
P e
+ y|
jx
topological semivector space over
a
the set of
,
with the Hausdorff metric topology.
R,
[a, b] ® [c, d] =
X[a, b] = [Xa, Xb]
K^(R)
Equip
.
^
U)
P
® q
(j.
U.
< X
P)
and that
9 XP
such
pP = XP
But this is
^ diam(— P)
is not continuous.
(\J.,
li
> diam(P).)
Thus, the topological
semivector space just considered, despite all its properties, does
not remain a topological semivector space when its topology is
strengthened in the fashion sought.
3.
CONVEXITY & POINTWISE CONVEXITY
The familiar notion of convexity for (topological) vector
spaces extends naturally and usefully to the case of topological
setnivector spaces.
In the latter spaces, a property which we term
"pointwise convexity" begins to assume an important role in its
This property, though automatically present in convex
own right.
vector spaces, needs to be assumed separately in the general case
of convex (topological semi vector) spaces.
In fact, it is when
the two properties (convexity and pointwise convexity) are combined
that they become specially useful.
3,0
Standing Notation
We denote the simplex {(X„, ..., X )
U
m
^
m+1
e R+
•••^= ^^
^y ^m
^"^ = °' ^'
iSo ^i
:
I
I
3,1
Definition
:
Let
be a convex topological semivector space
S
(i.e., a topological semivector space over
the usual real semifield).
their segment
e A
j.
[x:
A subset
whenever
x,
x'
T
Given any two points
is defined to be
x']
cz S
with
js
containing
x, x' e S,
= \x
will be called convex iff
® X'x'|
[x:
x']
ST,
Thus, what we call
a
convex topological semivector space
(see 1.0), indeed checks to be convex according to the above
definition.
(X,
X')
^T
The following are plain:
semivector space
e R)
(fi
=
if
;
then
S,
|a®b| aSA, bSBJ
((X,
|_lA
=
is convex in a topological
A
j^aj
is convex in
too,
B,
if
a
S,
e AJ
also is convex
then so are
®
A
B
Xa ® X
and all convex combinations
'B
e A^).
X')
It is important to note that, unlike in topological vector
spaces, in topological semivector spaces there is no guarantee
that
X
belongs to
x'
or
or even that
[x:x'J
x £ [x:x].
For example, give the discrete topology to the set
non-empty subsets
A,
B
C
R,
where
R
of all
[rj
stands for the set of reals,
and obtain a topological semivector space (with identity element
e =
A®B=
that
if
over the real field, with the understanding, as usual,
J0|)
X
|a+b| aSA,
and
?^
XA = R
be
otherwise
b},
(A,
while setting
B 6 [r];
Xa =
X e R)
space the only element belonging to its own segment is
{\a\
a e A
In this
.
Re
[rJ.
Possibilities such as the above motivate the following definition.
3.2
Definition
and let
|x|
3.3
CI
T
S.
T
be a convex topological semivector space,
will be said to be pointwise convex iff each
A convex topological semivector space
:
convex iff
[Prakash
C
S
is convex.
T
Exercise
Let
:
6e
(a + P)s = as ® Ps
Sertel, 1972].
for all
a,
P e
S
R_^
is pointwise
and
s
e S
10
Given a convex topological semivector space over a semifield
the largest pointwise convex subset of
®,
topological semivector subspace
semivector space over
i.e.,
,
forms a (convex)
S
topological
it is a
when considered under the restrictions
of the operations of
[Prakash & Sertel, 1972],
S
We close this section by defining the notion of convex hull
and recording two intuitively pleasing facts concerning convexity
in topological semivector spaces.
(See also the last sentence
preceding 5.1.)
3.4
Definition
T
and let
Let
:
c
be a convex topological semivector space,
S
The convex hull
S.
of
f
intersection of all convex subsets of
3.5
Exercise
Let
:
subsets of
e
S
by
T
J(T)
define (the
J(T),
®\t)l
m m
...,
(\„,
'
T
and
"
open simplex ")
mi
eA;
X)
m
T = UJCT(F)1 F e 3^(1)],
for each
and,
,
S
be as above.
if
containing
T.
Denote the set of finite
F = (t^,
a(F) =
...,
l(>^QtQ
t^)
«...
>0, i=0, ...,m}.
X.
S
is defined to be the
T
Then
is pointwise convex [Prakash
& Sertel, 1972].
3.6
Proposition
In convex topological semivector spaces,
:
topological closure (CI) preserves convexity.
Proof:
Let
Q
be convex in
S,
a
convex topological semivector
11
there is nothing to prove, so let
Q =
If
space.
adherent points of
(X,
X')
C1(Q).
e A^
Xq 9 X q' = q t C1(Q)
Suppose
Q.
Then there exists
.
The map
Q:
S x
S
being continuous, there is a nbd
n(U
X
u')
C
Since
V.
there exists
(y,
convexity of
Q,
q e C1(Q)
and
y')
Cl(y
,
C1(Q)
q
and
e U x
U'
y') e Q,
V
of
defined by
S,
-»
a nbd
U
q'
x
u'
is convex,
Vl(x,
of
(q,
for some
disjoint from
x') = Xx ® X'x',
q')
such that
are adherent points of
such that
a
q
y,
y' e Q.
contradiction.
as to be shown.
be
q
q,
Q,
Then, by
Hence,
12
4.
LOCAL CONVEXITY
Apart from preparation for their use in the fixed point theory
of Section 7, our motivation for stating the following "axioms of
local convexity" derives from the fact that, although for
topological subspace
X
a
of a (Hausdorff) topological vector space
the first three are always equivalent and all four are equivalent
when
X
is convex, we are able to assert only weaker relationships
between them in the case of topological semivector spaces.
a subset
in
X
Given
convex topological semivector space, we consider
a
the following alternative
4.1
Axioms
0.
;
For any
x € X
topology of
that
1.
U
c
V
there exists
a
x,
in the subspace
convex nbd
U
of
such
x
a
quasi-uniformity
S =
\e
<=
X
x
x|
a
£ AJ
of
inducing its subspace topology, such that, for each
E^ e
(S,
there exists
a
closed
cc
Eq e
c?
with
convex for each
There exists
a
x
E-
c
e
p
p
Ea(x)
2.
of
V.
There exists
X
X
and any nbd
e X.
quasi-uniformity
3 =
jE
CXxXlaeAJ
X
inducing its subspace topology, such that, for each
E
e S,
Eq(K)
there exists
a
and
a
closed
Eq e S
with
Ep
convex for each compact and convex subset
C
K
and
e
C
X.
of
,
13
a
e a}
each
of
X
e
(5,
E
=
c:
JE
x
x]
x
inducing its subspace topology, such that, for
there exists a convex
0° ll" 12° 13°
will be called
X
S
is convex and there exists a uniformity
X
3.
accordingly as it satisfies
E^.
&
with
S
locally convex (I.e. )
among these axioms.
0/1/2/3
Thus,
local convexity is the familiar local convexity.
4.2
Proposition
Given
:
subset
a
X
of
a
convex topological
semivector space,
(1)
If
X
is
1°
I.e.,
(2)
If
X
is
2°
I.e.
If
(3)
4.3
X
Proof
X
:
is
X
then it is
0°
be a
y / X,
y
In fact,
X
is
B =
T^
,
I.e.
jx|.
6
=
JB^^I
x e B = H B
Thus,
For, supposing
y
jt
B
Cu
a
,
As
x £ X.
at
e AJ
and
B
x
is
for some
y £ B
of
x
to which
for some
B
& 13
there exists a nbd
does not belong, whereby
is pointwise
I.e.
space, and suppose
I.e. T
there is a local base
I.e.,
as
2°
space which is
T^
consisting of convex nbds.
convex.
I.e.;
and pointwise convex, then it is
I.e.,
Every
:
Let
0°
3°
is
Proposition
0°
and
I.e.;
1
then it is
U
14
contradicting that
that
X
y e B.
Thus,
is pointwise convex,
jx}
is convex.
This shows
completing the proof.
i
Of course, all the local convexity properties
inherited by relative topologies on convex subsets.
0°-3°
are
In the next
section we turn to some basic facts relating local convexity
properties of Cartesian products with those of their factor spaces,
15
5.
CARTESIAN PRODUCTS OF TOPOLOGICAL SEMIVECTOR SPACES
Given
over
family
a
we equip
0,
jS
S
a
|
e
= H S
A "'
of topological semivector spaces
A;
with the product topology and define
its operations coordinatewise as follows:
where
ffi_
s
e
t
,
stands for the semigroup
of
o
r operation
r
a
are generic
S
(a e A)
Clearly,
.
topological semivector space over
jS
a
I
a
Evidently,
e a|.
a
c
S
=
S
11
^
convex iff each projection
X
=
c
(X)
TT
^
and
is then a
S
We call it the product of
0.
X
set
S
is convex/pointwise
a
is so.
S
a
5.1
Lemma
Let
:
|x
|
a
2°
be a family of
e a|
spaces of
I.e.
which all but finitely many are convex, and let
cS
quasi-uniformity inducing the product topology on
be a
X =
11
A
Then, for every
that
C
E
K = n K
A
Proof:
a
F
and
there exists
a
closed
is convex whenever
E(K)
of compact and convex subsets
Contained in
N
CA
of indices
F,
find a basic
K
K
a
cx
finite set
MCA
T € S,
H
£
c
tJ
E £ S
X
^
.
such
is the product
X_.
a
which restricts
of coordinates, including (w.l.g.) the set
m
H =
for which
X
nH n
(X
N
X
n
A\N
is not convex.
O.
xX),
a
Now
16
H
where
n
For each
n S N,
using the
such that
E
compact and convex
K
E
5,2
belongs to the quasi
-uniformity
^
'
n
e
Lemma
S
n
cz
n
n
The product of
:
H
n
ex.
n
a
2°
of
I.e.
with
S
X
x
N
family of
1°
(new).
n
find a closed
,
nE n
E =
X
convex for each
E (K )
n n
Write
n
of
n
(X
11
A\N
CC
1°
spaces is
I.e.
xX).
a
I.e.,
if all but a finite number of the factor spaces are convex.
Proof
5.3
Imitate the last proof.
:
Let
Exercise:
S =
be the product of a family of convex
S
11
A
°'
topological semivector spaces over
compact Hausdorff.
If the projection
^
-^
is Hausdorff,
5.4
Exercise
:
then
X
is
and let
0,
l°/2° I.e.,
X
a
=
rr
a
3°
I.e.
c
be
s
of
(X)
3°
into
X
accordingly as
The product of a family of spaces is
each of the factor spaces is
X
I.e.
X
is.
iff
S
a
17
6.
HYPERSPACES AS EXAMPLES
In this section we show some natural ways in which topological
semivector spaces arise as certain hyperspaces of topological
semivector spaces, e.g., of topological vector spaces.
In fact,
this is how, at first, we came to define topological semivector
spaces:
as an abstraction from hyperspaces of topological vector
spaces.
Our motivation derived from a need to be able to deal
with certain socio-economic adjustment processes in which some of
the variables were set-valued in topological vector spaces [see
Prakash, 1971; Sertel, 1971; Prakash & Sertel, 1971b].
This
abstraction not only enables one to eliminate cumbersome details
in the study of the above mentioned adjustment processes, but it
also turns out to afford many interesting examples besides those
discussed here [see Prakash & Sertel, 1972].
examples given here, however, form
a
The hyperspace
unified collection,
illustrating essentially all the salient aspects of (topological)
semivector spaces discussed in the previous sections.
In topologising hyperspaces, we use the upper semifinite,
finite or, when applicable, uniform topology, regarding all of
which we adopt E. Michael [l95l] as standard reference.
When
a
N.B.
:
topology is unmentioned, it is to be understood as discrete.
.
6.0
Standing Notation
Given
:
non-empty subsets of
0(X)
and
set
a
If
X.
will denote the set of
[x]
X,
is a topological space,
X
will denote the set of non-empty subsets of
?C(X)
which are closed, open and compact, respectively.
in a convex topological semivector space,
set of non-empty convex subsets of
C-^(X) =
=
6.1
KU)
(3-(X)
n i(X)
,
Let
:
=
JA
[xj
=
JA
e
..
.
,
= 1,
U
n
=
where
JA
e rxlt
-'
'
with
,
1
topological space
a
c:
A
(
1
.s.c.
[resp.
)
1
U U.
...,
U
Y,
a
)
,
A H U.
n
for
#
1
-L
-^
upper semi -continuous (u.s.c.
all collections of the form
[x]
'
U,
is an open subset in
U
open in
mapping
X.
f:
Y
Furthermore,
-»
[xJ
is called
[resp. lower semi -continuous
iff it is continuous with respect to the u.s.f.
l.s.f.
J
topology on
X.
is the one generated by taking as a
[x]
'-
..., nl
)
all collections of the form
[x]
[x]l A n U ^ 0},
>
all collections of the form
[x]
basis for open collections in
1
The upper semifinite
is the one generated by taking as a sub-basis
The finite topology on
given
?Ci(X)
and the lower semifinite (l.s.f.
u|,
for open collections in
i
and
is the one generated by taking as a
[x]
c
A
e [x]I
topology on
,
lies
Finally, we will denote
X.
be a topological space.
X
basis for open collections in
<U,
X
X
will denote the
.2.(X)
Oi(X) = OCX) n ^(X)
(u.s.f.) topology on
<U>"
If
,
n ^(x)
Definition
<U>
C-(X)
[xj.
(It follows that
f
is
,
19
continuous with the finite topology on
u
.
s
c
.
.
and
Let
1
.
s
c
.
.
)
be a semi vector space over
S
A©B={a®b|aeA, bSBJ
define
Then
(X e 0).
JZ(S)
Furthermore, if
Given
is convex,
S
then
^(S)
topological semivector space
semivector space and
S
into
K(S)
Furthermore,
K(S)
is
Section 2), i.e., a topology in which
0(S)
It
and
a
c: s
(3.2.(3)
a
is pointwise
is so.
S,
equip its hyperspaces
[s]
is then a topological
"
U ©
is a
If
S
strong " topology (cf.
s
is open,
is a topological semivector subspace of
[s]
is so.
S
(s
G S)
is open (such as in topological vector spaces)
follows that
subspaces of
^(S)
S
L^J*
S
Hausdorff iff
is a topological semivector space with a
then
iff
s,
e a(
and if
0,
CI
is embeddable as a topological semivector subspace
while
U
a
B
is a topological semivector subspace of
?C(S)
[s],
whenever
iXa|
A,
may be embedded as
S
with their respective finite topologies.
.
XA =
is pointwise convex only if
S
a
for any
is a semivector subspace of
topological semivector subspace into
convex; and
and
and,
is a semivector space over
[s]
then
is convex,
iff it is both
[x]
.2(5)
whenever
and
[s]
?C^(S)
[s].
are topological semivector
is a topological semivector space,
(topological) semivector space whenever
(topological) semivector space.
0(S)
is
.
20
6.2
Proposition
Let
:
with identity
be a convex topological semivector space
S
and let
e,
Assume that
convex.
X
(=
?C.2(X)
Hausdorff.
K-2-(X)
X
is pointwise convex),
0°
is
<V>
nbd
A
C
%<^KMX)
then
U
of
and
e
a
the
I.e.
is
?C.2(X)
W
of
local convexity of
Find
U
<=
x
is a nbd of
a
c
® W
while
V,
a
allows us to assume each
S
basic
a
there exist open
6 A
a
such that
a
V
Let
is so.
X
A.
Then
for each
®,
a
be a nbd of
with
?C^(X)
when
I.e.
<V>c5/-.
such that
A
By continuity of
X.
nbds
of
is
I.e.
and let
,
X
is pointwise convex (so that
X
the upper semifinite topology is
A e K-^-iX)
then
I.e.,
is
The rest being clear, we only prove that
:
yC^(X)
although it need not be Hausdorff even if
Furthermore, if
too,
,
only if
Proof
—
X
If
is
K-2-(X)
has a strong topology and equip
S
with the upper semifinite topology.
so is
Then
be convex.
s
to be
U
a
convex and the strong topology assures us that each
® W
U
open.
A,
JU
© W
a
I
thus being an open cover of the compact
e a}
it has a finite subcover
® W
\u
i
U = n U
T
a
and that
and
W
= U
T
U ® A
W
desired.
Denoting
ACU©ACU®WCV
•
Furthermore,
is convex.
A £ K-2.(X)
G l|.
i
|
i
we see that
,
a
<U
strong topology, so that
convex nbd of
is
a
a
,
®
A>
while
= K-^-iV
U ® A
®
A)
is open in the
is an open
<U®A><=<v>c:5/-,
as
21
6.3
Corollary
If
:
is convex in a
X
locally convex
(0°)
topological vector space (not necessarily Hausdorf f)
,
then
?C-2(X)
is convex, pointwise convex and, with the upper semifinite
topology,
Proof
as well.
I.e.
The topology of
:
linear topological space being strong,
a
the last proposition applies.
#
6.4
Corollary
be convex and
X
in a topological semivector
T
0°
space with strong topology.
Then
pointwise convex iff
with the upper semifinite topology
and
I.e.
is
Proof:
6.5
Let
:
X
?C.2(X)
X
is
Let
:
L
be a
(0
locally convex (not necessarily
)
uniform topology induced by that of
Let
:
(W_
OL
|
e A
each
|(x,
a
y)
X e L,
I
e A
and
ye
x £ L,
Then
L,
.2.(L)
is
,
I.e.
3
e
E^(x)
JE
\,
e L,
so that defining
where
E^(x) = x ® W^
CLxLJaeAJ
P € .2(L)
,
E
(P)
= P
® W
c=
L
is a nbd of
for
is a fundamental
system of entourages of the uniform structure of
any
with the
be a fundamental system of convex and
}
symmetric nbds of the identity
=
is
is pointwise convex.
Hausdorf f) topological vector space.
Eq^
?C.2.(X)
Use 4.3 and 6.2.
Proposition
Proof
I.e. and
Then, for
L.
P
c
L.
By
)
22
definition, the uniform structure on
a
|e
I
is the one generated by
e a1
J^(P) =
where
e
JT
i(L)
C
P
|
J
a
iff
and note that
Let
(P,
Q)
= Q
© XW
©
(P,
XV
Q
C
and
C
T
E^(P)
C
W^
Q ©
and
Q
C
c
© X'(Q' © W
a
P © W„.
X(Q ® W
W„ £ .2(L)
and
W
so that
a
P ® W^.
and consider an arbitrary convex
,
XP © X'p'
Xw^ © X'w
e AJ,
(P e ^(L)
!
so that
Since
a
|
To see this, fix
is convex.
P
^(L)
x
(\P ® X'P', XQ ® X'q'),
P =
.
J^
£
e <7
=
Q)
Now
.
convex, we have
Similarly,
Q)
(P'j q')
,
combination
P, Q e i(L)
(P,
a 2(L)
\J
E^(T)
It suffices to show that each
induced by
2.{L)
=
a
,
Hence,
)
is
.2(L)
P
c
)
pointwise
Q © W„.
ex
(P,
Q)
£
J„
,
so that
J
is
convex, as to be shown.
6.6
Corollary
:
Let
L
be as in 6.5.
uniform (equivalently
of
L,
Proof:
is
3
,
the finite)
6.5
,
,
K-l(L)
,
with the
topology induced by that
I.e.
(The parenthetically stated equivalence is directly
from [Michael, 1951, Theorem 3.3].)
.2(L)
Then
it inherits the
and is thus
3°
As
K^(h)
local convexity of
I.e.,
Jl(L)
is convex in
granted by
as to be shown.
#
.
.
23
6.7
Theorem
C
X
Given
:
be non-empty and convex, and equip
L
(equivalently
topology of
let
L,
with the finite
?C(X)
the uniform) topology induced by the subspace
,
KW
(so that
X
is closed in
Proof
Hausdorff topological vector space
a
Then
too, is Hausdorff).
,
K-^-OO
?C(X)
(The parenthetical assertions can be checked from [Michael,
:
1951, Theorems 3.3 and 4.9.3].)
Clearly,
KMx)
as
^ 0,
singleton sets are compact and, in a convex vector space, convex
as well.
Q e }(iX)
As
be a filterbase on
We show that
.
KU)
by that of
C
^
Let
= K(.L)
Q S K-2-(X)
n [x],
.
KW
the finite topology on
is the same as the subspace topology on
X
relative to the finite topology on
?(f(L)
converging to some
^J2.(X)
K(W
?C(X)
with
K(L)
As
-
induced
the finite topology is a topological semivector space (see the
comments following 6.1), the algebraic operations of
continuous.
by
fi(P)
convex,
For each
= XP
Q
© X P.
is into
(X,
X')
6 A
As each such
?C(X)
convex, the restriction
.
As
Cl
Fix attention to any particular
a nbd
of
liCL^iX)
Q(Q)
of
.
Q
By continuity of
such that
ClCU)
<^
?('.2.(X)
on
KW
f. K-^-iX)
(X,
X').
Q
-»
to
?('.2.(X)
Let
iTCiKiX)
there exists a nbd
"if.
As
-»
Q,
is
is,
?C.2.(X)
Cl,
/S
X
is pointwise
,
of each such
in fact, nothing but the identity map
Cl
are
is continuous and as
hence
X,
Q] ?C.2(X)
define a map
,
?C(L)
be
there exists an
24
^
element
n(y)
Q(Q)
e
such that
since
=5/-,
.
B
KW
As
6.8
.
c
:
Given
a
vector space
Denote
L,
B
let
n(^()
B
c
But
r.
converges to
and this is true for any
Q e ?('i(X)
,
X')
(X,
as to be shown.
locally convex Hausdorff topological
)
C
X
and
S
a
Q(y)
converges to a unique point.
is convex and
X
(0
by
?Co2.(L)
Then
^(.
This shows that
.
= Q,
fi(Q)
Q
Hence,
Exercise
?C^(X)
C
is Hausdorff,
This implies that
e A^
C
5/-
5/-
L
be non-empty, compact and convex.
KJl(X)
by
Equip
Y.
with the
S
finite topology and consider it as a (convex) topological
semivector space with operations induced in the usual fashion
by those of
Then
L.
having identity
0^
e_
is
S
=
space with identity
X
subset of
S.
Y
£
where
e S,
is a non-empty,
Finally,
L
x
S
is
pointwise convex, Hausdorff topological semivector
a corvex,
while
for each
Furthermore,
L.
I.e.
3
0£ = £
and
|0|
denotes the identity of
compact, convex and
Hausdorff and pointwise convex
X
Y e
e =
K-2.{L x
S)
(£,
e)
and is
and
3°
Ot = e
I.e.
for each
t
e L x
S,
26
game theory, economic theory and other instances of social analysis.
Given topological spaces
into the set of non-empty subsets of
upper semi-continuous (use)
given any nbd
that
C
f(U)
C
V
Y
of
,
and a mapping
Y
and
X
when we say that
Y,
we will mean that, for each
f (x)
there exists
,
a
nbd
C
we will write
C X D,
l(a, d)
7.1
Bx
I
n C
e B
THEOREM (Fixed Point)
semivector space
S
closed convex hull
X
)
n
e A
n
:
E
»
such that
(fixed) point
Given
Proof:
®
by
S
...
Let
® X
$(X,
a
cp:
n n
a
\l)
...,
a
=
g
c
A^ - X
I
of
such
X
$:
A
X
X
x
n
d)
G
e|.
A
-»
n
X
x
be the
X
...,
x
f:
X
-*
CJliY.)
Then there exists
.
cp(X)
= X a
be the map defined
Since the algebraic operations of
and
cp
F:
and
B
x
(x,
X = (X_,
x* £ f (x^'O
$.
be the graph of
be the graph of the map
A
'
be the map defined by
(cp(X), op(|a)).
X
and
and let
|
such that
e X
are continuous, so are
Let
x e X,
pointwise convex topological
'On' CS,
and let
,
cz
with Hausdorff topology, let
ja„,
x''''
F
e F
x)
(a,
be an upper semi-continuous transformation.
a
is
f
for the set (binary relation)
F
X = jX^a^ ® ... © X a
n n
of some
I
U
X
(This definition is equivalent to the one in 6.1.)
V.
For the composition of two binary relations
E
of
f
A
n
- [A
'-
n
f
]
and let
defined by
J
G
c
A
F(X)
'
\
n
x
A
n
27
=
continuity of
G = $"*"(§).
Thus,
(f(cp(\))).
cp
cp,
X = cp(A
g
is closed,
G
is closed, whereby
since
is closed,
since
y' = la^aQ ® ... ©
|Ii
=
PlJ.
+
[i
£ F(X)
,
y =
n
.
also, it
is continuous.
\s,
\l'
[i.e.,
e F(X)
y = Li„a„ © ... ® u a
U
n n
[J„ar.
u u
O,
P')
® ••• ®
yields
f(cp(X))
showing
A
F(X)
$,
cp
Ci
a
€ A
n n
and
define
,
Using
.
one may compute that
S,
and convexity of
Hence,
Let
For arbitrary
l-^^a^].
pointwise convexity of
e P'y'>
let
and denote
P'l-l',
is non-empty;
F(X)
is convex.
F(X)
y' £ f(cp(X)),
y,
,
is closed and
f((p(X))
We now check that
for some
X e A
Thus,
Hence, by continuity of
is use by compactness of
F
Clearly, for each
is compact, by
A
is compact, hence regular.
)
is use.
f
Since
y = Py
y £ f(cp(X)).
to be convex.
Now applying Kakutani's fixed point theorem, there exists
a
X* e A^
see that
such that
X-
£
Choosing
F(X*).
x* = cp(X*)
,
we
x* £ f(x^O.
#
7.2
Corollary
f:
X
-»
(
Kakutani's Fixed Point Theorem [l94l]):
C-.2.(X)
be an upper semicontinuous transformation of an
n-dimensional closed simplex
7.3
Let
X
c
into
R
exists a (fixed) point
x* £ X
such that
THEOREM (Fixed Point)
Let
X
:
f:
->
X
x^f
C-.2.(X)
.
£ f(x''0.
be a continuous
Then there
28
transformation of
subset
on
1°
(fixed) point
a
Since
:
I.e., non-empty, compact and convex
1
of a Hausdorff topological semivector space.
X
there exists
Proof
a
viy''
is compact,
X
we assume that
I.e.,
Since
cxxxjaeAJ
JE
.
there exists a unique uniformity
compatible with its subspace topology.
X
x* = fix*)
such that
6 X
Then
X
is
is a
fundamental system of closed entourages of this uniformity
such that
Define
E
(x)
Y„ =
jxl
is
(closed and) convex for all
We will show that
x 6 E„(f(x))j.
non-empty and closed for each
a
the theorem,
will imply
that
^ ^
X
for
x* €
fl.
CCSA
and
I.e.
X
,
cover of
X,
by
^0,
a
thus proving
'
x* = f(x*).
1°
and Hausdorff, hence
I.e.
let
JD
c
X
x
be a family of open symmetric entourages such that
Thus, for any given
C
Y
is non-empty,
Y
(aeA).
X
aeA
is pointwise convex (see Propositions 4.2
To show that
and ^.3).
e a}
T^
is
CO
Now first we note that, being
0°
Y
is non-empty,
Y 's
implies
Y^
e X.
Then, as the inter-
e A.
section of any finite collection of
of
compactness
^
x
n
U
ip
the map
F
city of
D
00 e... ®Xx
nn X=
on
E
x~,
U
..., x
|
n
x
e
e
X
Denote the closed convex hull of
= X-x„
C
{Dg^(^)
e A,
so that there exist
D (x.).
P =
a
I
P
,
by
(X-,
F„(p) = E (f(p))
for all
p £ P, F (p)
it is also closed and convex.
Thus
F
...,
C\
P.
a
D
c E^
is an open
XJ
with
|x_,
X)
eA
n
n
}.
,.., x
j
Define
Then, by symmetri-
is non-empty;
maps
x|
P
into
clearly,
(3-.2(P)
.
.
29
Denoting the graph of
E
<>
simply
^a ~
^^
*
of
in the compact
E
E
o
is use,
f
'^
Hence,
range
F
P,
Y
X
X
X - X
7.5
p e Y
(
X
x
is continuous,
regular (in fact,
showing that
,
is non-empty.
Y
since it is nothing but the projection
G
fl
where
A,
is the
A
This completes the proof.
X.
Tychonoff's Fixed Point Theorem [l935]):
Let
be a continuous transformation of a non-empty
such that
X
family of
1
of a locally convex linear
Then there exists
:
Let
JX
C
S
|
a
e a|
}f:X-»XaeAl
a'
a
Define
(fixed) point
be a non-empty
non-empty, compact and convex subsets of
I.e.,
Hausdorff topological semi vector spaces
.
a
x* = f (x*)
THEOREM (Fixed Point)
X = n X
is
(by the closedness
Thus, by Theorem 7.1, there exists
is use.
Hausdorff topological space.
X
X
as
is
is closed and, by compactness of the
compact and convex subset
e
f
F
Q*
Corollary
X*
is use
E
is closed,
CC
diagonal in
:
the graph of
,
and since
x)
of the compact set
n A)
(G
G
Since
x
G
T
i.e.,
,
'^
is obviously closed,
TT
f
^^
i.e.,
compact).
p e F (p)
7.4
Gq.
by
f
S
,
and let
be a family of continuous functions on
F:
exists a (fixed) point
X - X
x'"''
by
e
X
F(x) =
U^M \^Qf^-
such that
x-^
Then there
= F(x''0.
.
,
.
30
Proof
Clearly, the topological semivector space
:
=
S
11
S
A
is Hausdorff,
Since each
X
Furthermore,
C
X
and
1°
is
is non-empty,
S
so is
I.e.,
is continuous,
F
"
compact and convex.
(see Lemma 5.2).
X
as each
Hence,
is so.
f
the result follows readily by application of Theorem 7.3.
7.6
THEOREM (Minmax)
Let
:
1°
be
X
and
X
non-empty,
I.e.,
compact and convex subsets, each lying in a Hausdorff
Let
topological semivector space.
X = X
valued function on
f ^ (x„)
i
/
Max u(x
X
f
X
:
12
-X
XX
2
\
'
,
X
x^c)
=
such that
2
Min u(x
r^CY
zex^
and
f^:
z)
,
i
-X^.
X
Then
,
x„)
for all
12
,
(x
,
x
e X,
)
x„) S Max Min u(x
XX
12
12
,
x
)
)
F:
X - X
is continuous.
Min
=
'
that the function
x* e X
)
^
1
Clearly, the functions
f^Cx ))
Max u(y, x )
^
yex
12
X
x„) ^ Min Max u(x,
1
s Min u(x,
X
,
i
12
X,
=
^
x„) = Max Min u(x,
,
such that
,
x)
,
It is obvious that,
,
X
1
^
1
;
u(x
{x„l u(x
e
i
12
Min Max u(x,
Proof
I
1
define functions
2
jx
"^
f„(x,)
XX
e
x
be a continuous real-
u
uCx';!;,
f
and
f
defined by
F(x
,
x
)
=
(f (x
)
Then by Theorem 7.5, there exists an
x* = (x*, x*) = F(x*).
X ),
are continuous, so
Hence,
Max u(x
thus proving the desired equality.
,
|
31
7.7
THEOREM (Minmax)
Let
:
and
A
be non-empty but finite
A
sets, each lying in a pointwise convex Hausdorff topological
semi vector space, and let
hull of
and
A
A
12
=
)
define maps
f
Min Max u(x
XX
2
=
x
)
I
I
I
X = X
u(x
u(x
=
x
x)
2
,
1
'
- C-^CX
11x)
,
be the closed convex
X
XX
12
,
Let
be a continuous
u
such that
X-,
x)|
= Max u(y,
2
yex
Min u(x
=
zex
and
)
Max Min u(x
^
1
f
-
X
:
z)
,
i
(3-i(X
THEOREM (Fixed Point)
^
:
Let
X - C-.2(X)
f:
non-empty, compact and convex subset
topological semivector space into
(fixed) point
Proof
:
be an upper semi-
x* e X
x-'^
I.e.,
of a Hausdorff
X
C'.ZCX)
such that
2
Then there exists
.
e f (x*)
.
As in the proof of Theorem 7.3, it suffices to show
that the sets
Y
=
jx]
(f(x))|
x e E
closed, where, in this case,
'
IE
'
OC
a'
e A
system of closed entourages of the space
E (K)
Then
x ).
^
continuous transformation of a pointwise convex,
a
) .
Use Theorem 7.1.
Proof:
7.8
,
1
jx
X
:
^
jx
'
f (x )
^
I
and
respectively.
,
real-valued function on
f (x
X
is
are non-empty and
is a fundamental
1
X
such that
(closed and) convex for each non-empty, compact
.
32
and convex subset
K
CI
The proof is the same as that of
x.
Theorem 7.3, except that appeal is now made to the upper
semi-continuity, rather than the continuity, of
7.9
Corollary
(
Fan's Fixed Point Theorem [l952]):
f.
Let
X
be
non-empty, compact and convex in a locally convex Hausdorff
linear topological space, and let
semicontinuous transformation.
point
7.10
such that
X* e X
THEOREM (Fixed Point)
of pointwise convex,
Then there exists
jx
|
a
(fixed)
a
e A
be
j
non-empty family
a
non-empty, compact and convex
I.e.,
2
be an upper
-» C3-.2.(X)
x* S f (x*)
Let
:
X
f:
subsets of Hausdorff topological semivector spaces, and let
jf
:
X -
C-.2.(X
)
e A
CX
I
I
be a family of upper semicontinuous
X =
transformations, where
X
11
F(x) = n f„(x)
A
x''^
e
X
Proof:
ex).
(x
u
such that
Clearly,
Define
X
F:
-»
11
(3.2.(X^)
a
^
Then there exists
e F(x*)
x'^
.
ct
A
a
(fixed) point
.
is an use transformation of the pointwise
F
convex, non-empty, compact and convex Hausdorff space
^.^(X)
.
Although
convexity of each
fundamental system
that, whenever
K
by
X
X
need not be
I.e.,
the uniformity on
,
JE
1
|
i
e I
}
is the product
by
X
the
allows
X
1
into
local
a
of closed entourages such
K =
11
K
of compact and
33
K
convex subsets
a
and convex sets
=
jx|
,
cz
(x)
f
E
F(x)
is such a product of compact
X
(K)
Thus, as in Theorem 7.8, defining
.
is non-empty
Y.
implying that
e I,
i
.
i
it is clear that
X e E.(F(x))}5
and closed for each
is closed and convex (see
each
ct
Notice that
Lenma 5.1).
Y.
CX
Pi
Y.
and
^
proving the theorem.
7.11
THEOREM (Minmax)
2°
I.e.,
Let
:
and
X
be pointwise convex
X
non-empty, compact and convex, each lying in some
Hausdorff topological semivector space, and let
X = X
continuous real-valued function on
f
(x^)
12
f (x
2
define maps
respectively.
)
=
I
:
u(x
1
=
jx
I
X^
Then
,
,
1
-K3-.2.(X
)
Use Theorem 7.10.
X^
=
)
x)
=
and
Min u(x
f^:
,
1
x
)
2
such that
)
^
^ex
2
,
Max u(y, x
yex
Min Max u(x
X^
Proof:
12X
u(x
2'
1
f
|x
X
x
be a
u
,
z)
1
X^ - C-.^CX^)
=
,
Max Min u(x
X^
X^
,
1
x
)
2
34
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,
3.
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,
1952)
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,
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,
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,
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1951, pp. 152-182.
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35
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,
9.
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,
11.
(Prakash & Sertel, 1971b)
Prem Prakash and Murat R. Sertel, "Generalized Existence Theorems
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(Prakash & Sertel, 1972)
Prem Prakash and Murat R. Sertel, "Semi vector Spaces:
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Algebraic
(Schauder, 1930)
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Murat R. Sertel, Elements of Equilibrium Methods for Social
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,
15.
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A.
Tychonoff, "Ein Fixpunktsatz," Mathematische Annalen
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,
III
,
;7^