A note on ideals in topological quasi-vector spaces

              Rev. Acad. Canar. Cienc., XVI (Núms. 1-2), 143-155 (2004) (publicado en julio de 2005)
A NOTE ON IDEALS IN TOPOLOGICAL QUASI-VECTOR
SPACES
S. Jana, S. Mitra*
Department of Pure Mathematics
University of Calcutta
35, Ballygunge Circular Road, Kolkata-7000 19. India
e-mail address: sjpm12@yahoo.co.in
Abstract
In [2] we introd uced the concept of quasi-vector space asan associate of a vector
space. In this note we introduce the concept of ideal, maximal ideal, mínima! ideal in
a quasi-vector space and discuss their nature in sorne particular quasi-vector spaces.
Sorne aspects of ideals in topological quasi-vector spaces are also discussed.
AMS Classification: 46A99, 06F99, 15A99
Key words : Ideal, topological vector space, topological quasi-vector space.
1 Introduction
In a topological vector space the sum of two compact sets is compact; also scalar multiple of
a compact set is compact. So for a topological vector space X , if we consider the collection
C(X) of ali nonempty compact subsets of X then the aforesaid addition of two sets and
mul tipli cation of a set by a scalar become closed operations. Also the following results hold
for any two compact subsets A, B of X and any scalar a, (3
(i) A ~ B * aA ~ aB
' The second author is thankful to CSIR, INDIA for financia] assistance
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(ii) (o+,B)·A ~ aA+,BA. These simple facts have induced us to find a structural beauty in the
collection C(X) with the help of aforesaid addition, scalar multiplication and inherent set­inclusion
order. In doing so we have introduced in [2] a new structure which we call a "quasi­vector
space". lt has two structures, a semigroup structure and a partial order structure,
both being compatible with each other; also there is a scalar multiplication which generalises
the concept of vector space in the sense that every vector space can be embedded in a quasi­vector
space and every quasi-vector space contains a vector space as its subspace. fn this
space sorne properties of a vector space are lacking and sorne are present; the properties
which are absent appear in a different shape with the existing properties and with the
inherent order under consideration.
The study was carried out and a topology was introduced in this new space compatible
with its existing structures; a new topological algebraic structure was thus created which
was named "Topological Quasi-vector Space".
It has been observed that there is sorne novelty in the structure which is why there is an
endeavour to usher in further studies of such space.
In the present paper in §3 we have introduced the concept of 'ideals' in a quasi-vector
space, which is totally different from the well-known concept of ideal in a ring. In a ring,
ideal is a special type of subring whereas ideal in a topological quasi-vector space can never
be a sub-quasi-vector space, if it is to be a proper ideal. We have also defined a maximal
and minimal ideal of a quasi-vector space and shown that every quasi-vector space has a
unique maximal ideal; although a quasi-vector space rnay or rnay not have a rninirnal ideal.
We have also found a necessary and sufficient condit ion for a minimal ideal. Sorne simple
but useful results relating to ideal have also been obtained.
In §4 ideals in sorne particular exarnples have been discussed thoroughly.
In the last article, the topological character of an ideal has been discussed. The concept
of ideal has been utilised to find a necessary condition for a topological quasi-vector space to
be compact. In this context it should be noted that, a non-tri vial topological vector-space
can never be compact; but a topological quasi-vector space rnay or may not be cornpact.
2 Prerequisites
Definition 2.1 Let X be a nonernpty set and ':S:' be a partial order in it. Let '+' be a
binary operation on X and '·' : K x X ----> X be another composition [ K being the field
of complex nurnbers ]. lf ':S:', '+' and '·' satisfy the following axioms, we call (X,::::;,+,·) a
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quasi-vector space [ in short, QVS ].
A1 : (X,+) is a c01mnutative semigroup with identity '8'.
A2 : x :S y (x, y E X) =? x + z :S y+ z and a · x :S a· y, 't:fz E X , 't:fa E K.
A3 : (i) a · (x +y)= a · x +a · y
(ii) a· ({3 · x) = (a· {3) · x
(iii) (a + {3) . X::::: a . X+ {3 . X.
(iv) 1 · x = x, where ' l ' is the multiplicative identity in K, 't:fx, y E X , 't:fa, {3 E K
A4 : a . X = e jff a = o or X = e
A5 : x+(-l) ·x=8 iff xEXo= {xE X: YÍX, 't:fyEX\ {x}}
A6 : For each x E X , :J y E X0 such that y :S x.
Definition 2. 2 An element x E X is said to be oforder ' l ' if y í x, 't:fy E X\ { x}, otherwise
x will be said to ha ve order greater than 'l '.
Note 2.3 (i) Every qvs contains at least one ' l ' order element viz. the additive identity 8.
(ii) One order elements are the only invertible elements in X. In-fact: a+ b = B(a, b E
X) ==> a+ b - b = -b +e = -b. Again, a+ b - b 2:: a+ e ==> -b 2:: a==> b 2:: - a ==> 8 =
a+ b;::: a - a;::: e==> a must be of order ' l ' ( by axiom A5 ). Similarly b must be of order ·
' l '. It now follows that, a + b (a , b E X) will be of order ' l 'iff both a and b are of order 'l'.
(iii) We also observe that for each m E X0 , :J y E X s.t y ;::: m and y -=f. m for, if x be an
element of order greater than 'l' then X - X+ m 2:: m and X - X+ m -=j. m [ since X - X -=j. 8 ].
Definition 2.4 [l] A partía! order ':S' in a topological space Z is said to be closed if its
graph {(x,y) E Z x Z : x :S y} is closed in Z x Z, endowed with the product topology.
Theorem 2.5 [l] A partial order ':S 'in a topological space Z will be a closed order iff for any
x, y E Z with x í y, :J open nbds U, V of x, y respectively in Z such that (j U) n (l V) = <P,
where, j U = { x E Z : x ;::: u for some u E U} and l V = { x E Z : x :S v f or some v E V}.
Definition 2.6 A qvs X is said to be a topological qvs if X has a topological structure with
respect to which '+' and '·' are continuous and ':S' is a closed order su ch that for each open
set V in X, l V = {y E X: y :S x for sorne x E V} is open in X.
Proposition 2. 7 Every topological qvs is Hausdorff.
Proof : Let X be a topological qvs. Let x, y E X with x -=f. y. Then either x í y or y í x.
Without loss of generality let, x í y. Since the order ':S' is closed, :J two open nbds U, V of
x, y respectively in X such that (i U) n (l V) = <P [ by theorem 2.5 ] => Un V = <P.
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3 Ideals in a QVS
Definition 3.1 A nonempty subset l of a qvs X , is said to be an ideal of X if
i) X+ l ~ l ii) a.I ~ l , a-=/- O iii) j l = l where j l = {x E X: x 2: y for sorne y E!}.
Note 3.2 We observe that if e E l then 1 = X; therefore l will be proper ideal of X iff
e fj_ l. It is equivalent to say that l cannot contain any element of order '1'. Thus if X 0
denotes the set of ali '1' order elements of X, then l is a proper ideal of X iff l n X0 = <!).
Theorem 3.3 X\ X0 is a proper ideal of X.
Proof: First we have X0 -=f- X. Let x E X and a E X\ X0 . Then a cannot be of order ·1'.
We claim that x+a is also not of order '1', for otherwise, (x+a)-(x+a) = x-x t-a-a = e.
Again, X - X 2: e =? a - a + X - X 2: a - a =? e 2: a - a =? a - a = e =? a is of order '1'
which is a contradiction.
Now, Jet a E K , a-=/- O and a E X\ X0 . We show that aa E X\ X0 . For, if aa E X0 then
aa - aa = e=? a(a - a) = e=? a - a= e (a-=/- O) =?a E X0 which is a contradiction.
Let x E j (X \ X0 ). Then 3 a E X\ X0 s.t x 2: a. We show that x E X \ X0 . If not,
X E Xo =?X - X= e. Thus, X;::: a=? X - X;::: a - a=? e;::: a - a=? a - a = e=? a E Xo
which is a contradiction.
Consequently j (X\ X0 ) = X\ X0 [ since (X\ X0 ) ~ j (X\ X0 ) ] . Therefore X\ X0 is an
ideal of X. Clearly it is a proper ideal since X0 -=/- <t>.
N ate 3.4 We note that X\ X0 is a maximal ideal of X in the sense that, there is no proper
ideal l of X such that X\ X0 e l. It is also clear that X cannot have any other m&'<.imal
ideal i.e. X \ X0 is the unique maximal ideal.
Theorem 3.5 Arbitrary non empty intersection of ideals is an ideal.
Proof: Let {Ja: a E A} be an arbitrary family of ideals of X such that íla EA la-=/- <t>.
Let l = ílaEA la
i) Let a E 1 then a E la Va E A
Therefore, x +a E la Va E A [ since la is an ideal ]=? x +a E ílaEA la= 1 Vx E X.
ii) Let a E l. Then a E la Va E A
Therefore, {3a E la for ali non zero {3 and Va E A [ since la is an ideal ]=? {3a E ílaEA la =
l V{3-=/- O
iii) Let p E j l.
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Then p ~ q for sorne q E J =? q E J0 Va E A =? p E i J0 Va E A
=? p E J0 Va E A [ since J0 is an ideal of X, i J0 = J0 ].
Therefore, p E íla EA la = I.
Consequently, i I = I [ since I ~ i I ]
This completes the proof.
Proposition 3.6 Arbitrary union o/ ideals o/ X is an ideal.
Proof: Immediate from defini tion.
Theorem 3. 7 Let a E X, where X is a quasi-vector space. Then J(a) = {y E X : y ~
x + aa, x E X , a E K*} is an ideal o/ X containing a, where ]{* = J{ \ {O} .
Proof: Clearly, a E J(a) [ since a= 8 + l.a where 8 E X , 1 E K* ].
i) Let x1 E X and y1 E J(a) Then :J x2 E X and a E K * such that y1 ~ X2 + aa
Now, x1 + y1 ~ x1 + x2 + aa =? x1 + Y1 E J(a) [ since X¡+ X2 E X]
Therefore X + I(a) ~ I(a)
ii) Let a E J{* and y E J(a) . Then :J x E X and (3 E K * such that y~ x +(Ja
=?ay~ a(x +(Ja) = ax+ a(Ja =? ay E J(a) [ since ax E X and a(J E K* ]
=? a I (a) ~ I(a)
iii) Let p E i J(a). Then :J q E J(a) such that p ~ q
Now, q E J(a) =? :J x E X and a E K * such that q ~ x + aa
Therefore p ~ q ~ x + aa =? p E J(a) =>i I(a) = I(a) .
In view of (i), (ii) and (iii) t he theorem follows.
Corollary 3.8 If 'a' be an element o/ arder '1 ', then I (a) = X and conversely.
Proof: Let x E X. Then there exists an element y in X of arder '1' such that x ~ y=?
x ~ y - a + a [ sin ce a - a = B for, a is an element of arder '1' ] .
Now y - a E X=? x E J(a) . Therefore X = J(a).
Conversely, let I(a) = X for sorne a E X. Then BE J(a). So :J x E X and a E K* such that
8 ~X+ aa =?X+ aa = 8 =? a-1x +a = 8 =?a is of arder '1' [ by note 2.3 ].
Note 3.9 The ideal I(a) is said to be the ideal generated by 'a'. It is the smallest ideal
containing a; for, if J be another ideal containing a such that J ~ J(a) then p E J(a) =? p ~
x + aa for sorne x E X, a E K*
Now, x + aa E J [ since J is an ideal anda E J] ==? p E i J = J.
Thus, I(a) ~ J =? I(a) = J.
It is also clear from theorem 3.5 that I(a) is the intersection of ali ideals containing a.
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Theorem 3.10 Any two proper ideals must intersect.
Proof : Let I and J be any two proper ideals of X. Now, I + J ~ J and J + I ~ J.
Consequently I + J ~ J n I
Theorem 3.11 For any ideal J ~ X, l J = X, where l J = {x E X : x ::::; j for some
j E J}.
Proof : Let x E X. Then either x E J or x rf- J.
Now, x E J =? x E l J =? X ~l ] =? l J = X.
Let x rf- J. Now I(x) n J f <P [ by theorem 3.10 ] =? 3 p E X such that p E I(x) and
p E J =? 3 X ¡ E X and a E K* such that p ~X ¡ +ax=? p-X¡ ~X¡ -X¡ +ax~ e+ax = ax.
or a-1(p - x1) ~ x [ since a f O]
Now, a-1(p- x1) E J [ since J is an ideal of X and p E J]
Therefore there exists an element a-1(p- x1) E J such that a-1(p- x1) ~ x =? x El J ==;.
X~ l J. Therefore l J = X.
Proposition 3.12 If a::::; b (a , b E X) then I(b) ~ I(a)
Proof : Straight forward.
Theorem 3.13 For any ideal J ~X , J = UxEJ I(x).
Proof: Let x E J =? x E I(x) ~ UxEJ I( x).
Conversely, !et y E UxEJ I(x). Then 3 p E J such that y E J(p) =? 3 a E X and a E K*
such that y ~ a + ap.
Now, a+ ap E J [ since p E J and J is an ideal ].
Therefore, y E J [ since j J = J] =? UxEJ I(x) ~J. Thus UxEJ I(x) = J.
Definition 3.14 An ideal I of a qvs X is said to be a mínima! ideal of X if there <loes not
exist any proper ideal J of X such that J ~ J.
Theorem 3.15 J is a minimal ideal of X iff J = I(x) \fx E J.
Proof: Let x E J. Then I(x) ~ J [by note 3.9 ]. Also, J being minimal J ~ I(x).
Conversely, Jet J = I(x) \fx E J. If possible, !et 11 be another ideal of X such that ] 1 ~J.
Let p E 11 =? p E J =? J = I(p) . Now, I(p) is the smallest ideal containing p.
Hence I(p) ~ 11 =? J ~ 11. Consequently J is minimal.
Note 3.16 Since any two ideal intersect, mínima! ideal (if any) of a qvs must be unique.
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Theorem 3.17 If a qvs X contains a maximal element a, w.r.t the partial order then I(a)
is a minimal ideal.
Proof : Let x E I(a). Then I(x) S: I(a). Also x 2 y+ cw for sorne y E X , a E J{* =?
x - y 2 aa =? a-1(x - y) 2 a=? a-1x - a-1y = a [ since a is a maximal element in X]
=?a E I(x) =? I (a) s;; I(x ).
Theorem 3.18 Let A be a subset of a qvs X. Then
I(A) = {y: y 2 x + 2=?= 1 aiai, ai E K* , ai E A, i = 1, .. . ,n;x E X ,n = 1, 2, 3, ... } is an
ideal of X containing A.
Proof: Let a E A. Then a = B + 1.a =?a E I(A) =? A S: I(A). The rest of the proof is
similar to that of theorem 3. 7.
Note 3.19 Arguing similarly as in note 3.9 we can say I(A) is the smallest ideal containing
A.
Proposition 3.20 I(A) = UaEA I (a).
Proof : Let b E I(A). Then :3 x E X and a1 , ... , an E A, a 1 , ... , O:n E K* such that
b 2 x + 2::?=1 aiai = x + I:?==-/ aiai + O:nan . Therefore, b E I(an) [ since x + I:~/ a;ai E X
]=> b E UaEAI(a). Let p be an element in UaEAI(a) . Then :3 a E A such that p E I(a) .
Therefore p 2 x + aa for sorne x E X and a E K * =? p E I(A).
Therefore, I(A) = UaEA I (a) .
4 Examples of ideals
4.1 Example
Let X be a topological vector space over the field of complex number K. Let C(X) be the
set of all non-empty compact subsets of X. We define addition ( +) and scalar multiplication
(-) in C(X) as follows:
Let A,B E C(X) and a E K
A+B= {a+b:aEA,bE B} , a· A = {aa:aEA}.
Clearly A + E and aA are compact since the addition and scalar multipli cation in X are
continuous, X being a topological vectorspace and hence A+ E , a · A E C(X). It is easy to
check that C(X) with the aforesaid operations and with respect to the usual set-inclusion
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order forms a qvs over K , {8} being the identity element in C(X ), where 8 is the identity
in X.
Ideals of C(R)
Let us consider the topological vector-space R. The closed and bounded subsets of R are
the only compact subsets of R. It is easy to observe that the singleton sets {a}, a E R are
the only one order elernents of C(R) . Also C(R) \ S is the rnaxirnal ideal of C(R), where
S = {{a}: a E R} ( by note 3.4).
We now discuss the ideals generated by single elements of C(R).
Proposition 4.1.1 Let A,B be two two-point sets of R. Then I(A) = I(B).
Proof: Let A = {a1,a2 } , B = {b1, b2}. Then I(A) ~ I (B) iff A E J(B) iff A :2 E +aB for
sorne E E C(R) and a E R \ {O} = R* (say)
Since A and B contains same number of elernents it follows that E must be singleton and
hence A = E+ aB . ( *). Let E = { x}. Then above relation (*) holds iff a 1 = x + ab1 and
a2 = x + ab2 (if a1 = x + ab2 and a2 = x + ab1 then B can be renamed just by interchanging
b1 and b2 so that we get the above relations) which being a systern of two linear equations
in two variables x and a must have a unique solution.
Consequently, I(A) ~ I(B) is always true.
Sirnilarly, J(B) ~ I(A).
Proposition 4.1.2 If A = {a1, ... , an}, B = {b1, . .. , bn} where n > 2, then I(A) = I (B)
1 a; b;
iff ai bi = 0 for15:, i, j , k5:,n, i-=fj-=fk.
ak bk
Proof: I(A) ~ I(B) iff A E J(B) iff A :2 E+ aB for sorne E E C(R) and a E R*.
Since A and B contains sarne nurnber of elernents it follows that E rnust be singleton and
hence A = E+ aB. Let E = {x}, x E R. Therefore A = {x} + aB. This is true iff the
following system of linear equations a; = x +a bi, i = 1, . .. , n in x and a has a solution.
If a; = x + abi, i , j E {l, 2, ... , n} instead of the said equations, then B can be renarned in
such a way that we get the above equation(s. 1 bb.ni. J
Here arder of the coefficient matrix P = is n x 2 and that of the augmented
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matcix Q ~ ( : . :: : ) irn x 3 Theabov"y'tem hes ""ºI ot ion i ff '( P) ~ '( Q),
where r(P) denotes the rank of P. In this case, r(P) :S 2 and r(Q) :S 3. So if r(P) = r(Q)
1 b; a;
then r (Q) f:. 3 and hence every minor of Q of order 3 is zero i.e bJ ªJ = O for
bk ak
1 a; bi
1 :.:::; i, j , k :.:::; n ; i f:. j f:. k ==?- ªJ bJ = O for 1 :S i, j , k :S n, i f:. j f:. k ..
ak bk
1 b
Conversely, if these deterrninants be all zero then r(Q) f:. 3. Also ' f:. O 'í!i ,j(i f:. j)
1 bj
[ since b; f:. bJ, 'í!i, j (i f:. j) ]. So r(P) = r(Q) = 2. Consequently, the above system of n
linear equations in two variables x and a has unique solut ion. If we interchange A and E in
above arguments then, under same condition, I (B) <:::: I(A) . This completes the proof.
Note 4.1.3 lt is easy to note that if A contains n elements and E contains n+ 1 elements
of R, then I(A) cannot be contained in I(B). Also I (B) is not necessarily contained in I(A).
Proposition 4.1.4 If A = {a1, . .. , an}, E = {b1, ... , bn+i} where n > 2, then I (B) e I (A)
1 a; br
iff ªJ bs = O f or 1 :S i, j , k :S n ; 1 :S r, s, t :S n + 1; i f:. j f:. k and r f:. s f:. t.
1 ak bt
Proof : Similar as proposition 4. 1.2
Note 4.1.5 (i) We observe that any ideal generated by a countably infini te compact subset
of R is contained in sorne ideal generated by a finit e subset of R. Also, any two ideals
generated by two countably infinite compact subsets of R need not be same. For example,
I( { l /p: p is prime} U {O}) and J( {1 /n : n = 1, 2, 3, ... } U {O} ).
(ii) Each ideal generated by countably infinite compact subset of R contains an ideal gen­erated
by sorne uncountable compact subsets of R. Also any two ideals generated by two
uncountable compact subsets of R need not be same. For example, J( {2:::~ 1 r,. : a; = O, 2})
and J( {I:~ 1 r, : a;= O, 2, 4} ).
Minimal Ideal Of C(R) .
The closed interval [O, l ] is an uncountable compact subset of R. We claim that J([O, 1])
is contained in any ideal of C(R). In fac t, we show J ([O, 1]) <:::: I (A) for any compact subset
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A of R. For this we are only to show that [O, 1] E J(A). Since A is a compact subset of R,
:J a,b E R such that A~ [a,b] ~ {b-:=.:} + b~a A ~ {b-:=_:} + b~a[a,b] = [0,1] ~ [0, 1] E J(A).
We now prove that J([O, 1]) is the minimal ideal of C(R). Let E E J([O, 1]). Then obviously
I(B) ~ J([O, 1]). Also by above discussion, J([O, 1]) ~ I (B) since E is a compact subset of
R. Thus J([O, 1]) = I(B) 't:/B E J( [O, 1]). Therefore by theorem 3.15, J([O, 1]) is the minimal
ideal of C(R).
4.2 Example
Let G be the set of ali 2 x 2 matrices A over R such that IAI ;:::: 1, where IAI denotes the
determinant of A. We define a relation p on G as follows :
ApB holds iff IAI = IBI. Then clearly p isan equivalence relation on G. This determines
a partition on G. Let [A] denotes the equivalence class containing A.
Let X = {[A] : A E G}. We define addition ( + ), scalar multiplication (·) anda partial order
'<' on X as follows :
(~ [AJ + [BJ ~ [ABJ (ji) a[AJ ~ [aAJ.a j O and O[AJ ~ [JJ whece J ~ ( ~ : )
(iii) [A] :::; [E] iff IAI :::; IBI. It is easy to verify that ali operations defined above are well
defined.
Then (X,:::;,+, ·) is a qvs over the field R.
Ideal of X
Here [J] is the only one order element, i ,e. X0 = { [J]}. So, X\ { [J]} is the maximal ideal
of X.
Let [A] E X\X0 . Then, J([A]) = {[E] E X: IBI 2: IAI}. Hence, J([A]) = J([B]) iff IAI = IBI.
Also, for any two elements [A], [E] E X\ X0 either J([A]) ~ J([B]) or J([B]) ~ J([A]). Thus
the collection of ali ideals of X forms a chain of ideals. We claim that X has no mínima!
ideal.
If possible let, J is the mínima! ideal of X. Then :J [A] E X\ X0 such thát J = J([A]). Let
E be a 2 x 2 square matrix over R such that IBI > IAI- Then [E] E J([A]) = J ~ J([B]) C J
but J([B]) =f. J [ since IBI =f. IAI ]. This contradicts the minimality of J.
4.3 Example
Let (S, :::;) be a lattice with the least element e. We define* : s X s---+ s by a* b = lub(a, b)
where lub(a, b) denotes the least upper bound of a, b. Also We define ' ·':K x S ---+ S by
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{
a,
a.a = e,
Ideal Of S
for a=/- O
for ali a. Then (S, *, · , ~) is a qvs over K (K being a field ).
for a= O
Here e is the only one order elernent of S.
Proposition 4.3.1 Let e=/- a E S. Then I(a) = i a
Proof: Let p E ja=;. p;::: a=;. p;::: 8 * aa 'Va =f- O=;. p E J(a)
Conversely Jet, q E I(a) =;. q ;::: x * aa for sorne x E S and a E K \{O} =;. q ;::: x *a [ since
aa = a for non zero a ] . Let l = x * a then l ;::: a. Therefore q ;::: l ;::: a =;. q E j a.
Proposition 4.3.2 I(a) = J(b) iff a= b
Proof: Follows irnrnediately frorn proposition 4.3.1
Proposition 4.3.3 S has a minimal ideal iff it has a maximal element.
Proof: Let a be a rnaxirnal elernent of S. Then I(a) = ja = {a} =;. I(a) is a minimal
ideal.
Conversely, let J be a minimal ideal of S. Then J = I(a) for some a E S.
Now, x ;::: a(x E S) =;. x E j a = J =;. J = I(x) [ since J is the rninimal ideal ]=;. I(x) =
J(a) =;. x = a ( by proposition 4.3.2). So a is a rnaxirnal elernent of S.
5 Topological property of ideal in a topological qvs
Theorem 5.1 Let I be an ideal of a topological qvs X. Then I is also an ideal of X.
Proof: i) Let x E X and p E J. Let V be any open nbd of x +p. Then :Jopen nbds W1
and W2 of x and p respectively in X such that W1 + W2 i;;; V.
Now, W2 is an open nbd of p and p E I =;. W2 n I =f- <I>.
Let q E W2 n J. Then x + q E V [ since q E W2]
Also, x + q E X+ J i;;; J [ since q E J and J is an ideal]=;. x + q E V n J =;. V n J =f- <I>.
Hence, x + p E J. Thus, x + p E X + I =;. x + p E l. Therefore, X + I i;;; l.
ii) Let a E K* ,p E I and V be any open nbd of ap. Then :Jopen nbds N of a in K and U
of p in X such that NU i;;; V.
Therefore, Un I =/- <I> [ since p E l and U is an open nbd of pin X ]. Let t E Un J. Then
at E I [ since t E I and I is an ideal ]
Also, at E V [ since a E N, t E U and NU i;;; V]=;. V n I =f- <I> =;. ap E I
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iii) Let, q E j l. Therefore, :3 p E l such that q ;:::: p.
Let W be any open nbd of q. So, p E l W [ since q ;:::: p ]. Now l W being open in X ,
l W n J =fe <I>. Let z E l W n J. Therefore, z '.S: w for sorne w E W and z E J =? w E j J = J
[ since J is an ideal of X ]=? W n J =fe <I> =? q E l =? j l s.;; l. Consequently j l = l [ since
l s.;; i l ].
In view of (i), (ii) and (iii) the theorem follows.
Definition 5.2 An ideal J is said to be a closed mínima! ideal if J is closed and it contains
no closed ideal properly.
Proposition 5.3 If J be the minimal ideal of a topological qvs X , then J will be its closed
minimal ideal.
Proof: If possible !et J be a closed ideal such that J s.;; J. Since J is the mínima! ideal,
J s.;; J s.;; J =? J = l = J. So J is a closed minimal ideal.
Theorem 5.4 J(A) is path-connected jor each A (s.;; X\ X0 ).
Proof: I(A) = U aEA J(a) [by proposition 3.20]. We first show that I(a) is path connected
for each a E A. Let y1, y2 E I(a). Then :3 x1, x2 E X and o:1 , o:2 E K* such that y1 :2'. x1 +o:1a
and y2 :2'. x2 + o:2a. We define a fun ction f : [O , 1] ->X by f (t) = (1- t)y1 +ty2. Clearly f is
continuous. Also, j(t) = (l-t)y1 +ty2 ;:::: (1-t)x1 +tx2+(1-t)o:1a+to:2a =? f(t) E I(a) Vt E
[O,l]. Thus f is a continuous path in I(a) such that f( O) = y1 and j(l) = y2 . Consequently
I(a) is path-connected for each a E A. Since any two ideals intersect it follows that I(A) is
also path-connected.
Theorem 5.5 If X be a compact topological qvs1 then it has a minimal ideal which is also
compact.
Proof: We first show that X has a closed mínima! ideal. Let r be the collection of ali closed
ideals of X. Since closure of an ideal is also an ideal ( by theorem 5.1 ) r is non-empty.
Let J = ílFH F. Since any two ideals intersect (by theorem 3.10) r has finite intersection
property. Since X is compact, J =fe <I>. So J must be a closed ideal ( by theorem 3.5 ). Also
from the construction it follows that J must be the closed minimal ideal.
We now prove that J is mínima!. For this we have to prove J_ = I(a) Va E J ( by theorem
3. 15 ).
Let a E J. Then I(a) s.;; J =? I(a) = J [ since J is closed rninimal ideal ]. If we can show
that I(a) is closed the proof is done.
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Let {Yn}nED be a net in I(a) converging to y, D being a directed set . So for each n E
D :J Xn E X and CYn E K* such that Yn :'.'.: Xn + CYna --~ * Since X is compact, the net
{xn}nED in X has a convergent subnet {xm}mEE(say) , E being a directed set. Let Xm-+ x.
From * we have Ym :'.'.: Xm +ama, m E E. Here { am}mEE is a net in K*. Now two cases may
arise.
Case-I : {am}mEE has a bounded subnet {ap}pEE' (say), E' being a directed set. Then
{ CYp }pEE' has a convergent subnet { ªº} qED' (say), D' being a directed set. Let ªº -+ a =;.
ªºª-+ aa [ since '" is continuous ]. Again J is closed and CYqa E J \:Jq E D' =;. aa E Ji.e.
a =f. o [ since e =f. J l
Now we have yq :'.'.: Xq + ªºª =?y :'.'.: x + aa [ since the partial order ':::;' is closed and any
subnet of a convergent net is convergent and converges to the same linút, limit being unique
for , X is Hausdorff] where a =f. O=;. y E I(a).
Case-II : { CYm }mEE has no bounded subnet. Then { CYm -l }mEE is a bounded net in K* and
hence it has a convergent subnet {ap- 1 }pEEi (say), E1 being a directed set. Let ap - 1 -+ (3.
Now { apa }pEEi is a net in J and J is compact [ since J is closed subset of X which is
compact ]. So {apa}pEEi has a convergent subnet {aqa }qEE2 (say), E2 being a directed set.
Let ªºª -+ z. Then z E J. Again , ªº -1 -+ (3. So ªº -1ªºª -+ f3z =;. a = (Jz [ since limit of
a convergent net is unique ]=;. f3 =f. O [ since a =f. e]=;. z = (3 - 1a. Thus ªºª-+ f3 - 1a where
(3-1 =f. O. Again we have yq :'.'.: Xq + ªºª \:Jq E E2 =? y :'.'.: x + (3- 1a [by same logic as in case-I ]
=;.y E I(a).
Consequently, I (a) is closed.
Now J being a closed ideal it must be compact, since X is compact.
References
[l] Leopoldo Nachbin (1965); Topology And Order, D.Van Nostrand Company, Inc.
[2] S.Ganguly, S.Mitra and S.Jana; An Associated Structure Of A Topological Vector Space;
Bull.Cal.Math.Soc. ,96,(6) 489-498(2004)
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