Thesis Abstract and Dissertation Abstracts
Some Of Study On Cone And Topological Space
Abstract Category: Other Categories
Course / Degree: PhD
Institution / University: University of Allahabad, India
Published in: 2014
Abstract
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.
More precisely, it is the solid figure bounded by a base in a plane and by a surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base, such that there is a circular cross section. The term "cone" sometimes refers just to the surface of this solid figure, or just to the lateral surface.
The axis of a cone is the straight line (if any), passing through the apex, about which the base has a rotational symmetry.
In common usage in elementary geometry, cones are assumed to be right circular, where right means that the axis passes through the centre of the base (suitably defined) at right angles to its plane, and circular means that the base is a circle. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base.[1] In general, however, the base may be any shape that permits a circular cross section of the cone, and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite area, and that the apex lies outside the plane of the base).
INTRODUCTION
In this work we can define ourselves with the study of Cone for the definition of which, we refer to page 176 of Topological Vector Space and Distribution Volume-1 by John Horvath Addison Wesley Publishing Company (1966). Also deals with Bounded Subset in topological group.
The whole has been divided into four chapters. In chapter — II, We have defined a new set called M-Cone in a linear space. It is more general than a convex subset is less general than a cone. A M-Cone has been defined as follows.
In Chapter-Ill. We have defined the theorems and bounded subset in topological group.
We have shown that R1 is a closed ideal of R.
The chapters are divided into sections, and equations in which we will occasion to refer are numbered consecutively in each section (2.5) means section 5 in chapter 2. Theorems arc denoted by a section number followed by Roman numerals, the whole being placed in brackets, thus (25V) refers to the fifth theorem in the section (2.5).
M-CONE
2.1 INTRODUCTION
In this chapter we have studied M-Cones in a real or complex linear space. We have defined M-Linear hull of a set in a linear space and studied some of its properties. We have studied the product of two M-Cones. We have studied the properties of M-Cone of a sequence in the complex plane and studied its properties. We have defined M-Conical functional as well as IC- Convex junction and studied them. We have also defined and studied a linear space S satisfying ascending chain condition (ACC) and maximum condition on M-Cones. Finally we have defined B : A and D, (A) and B being subsets of a linear space and studied with reference to M-Cones.
2.2 M-CONE
Let A be a non-empty subset of a linear space L for x, y A, ax + βγ
A, Where a ≥ 0, β ≥ 0 and a + β ≤ 2, then we define A as a M-Cone.
Theorem 2.2.1
The intersection of a family of M-Cones in a linear space L is also a M Cone.
Theorem 2.2.2
Let { Ai } be falmily of M-Cones such that Ai A2 A3 A4 ..........
that is Ai Aj Whenever i ≤ j, then Ai. is also a M-Cone.
Theorem 2.2.3
Any finite linear combination of any M-Cone is again a M-Cone.
Theorem 2.2.4
A family A = {Ai} of all M-cones of a linear space L which are totally ordered by set inclusion, is a complete lattice.
Theorem 2.2.5
Every linear space has a maximal M-cone.
Theorem 2.2,6. Let T be a linear transformation from the linear space X to the Linear space Y. Then the image of each Art-cone in X is a M-cone in Y and the inverse image of each M-cone in Y is a M-cone in X.
Theorem 2.2.7
Let A be M-Cone and x1, x2, x3............xn A. n being a positive integer.
If a1, a2, a3.........an are n given scalars such that
0 ≤ ai ≤ 2 and ai ≤ 2, then ai xi A
2.3. M-LINEAR HULL
Let A be any set in a linear space L, then an expression of the form tl x1 +
t2 x2 + ........+ tn xn in which 0 ≤ ti ≤ 2 for all i = 1, 2,3 ...... n, ti 2 and xi A, , n being any positive integer, is called a M-Linear combination of elements of A, The M-Linear hull of A in short denoted by M(A) is the set of all M-Linear combinations of the element of A.
Theorem 2.3.1
If A and B are subset of linear space L such that
A B, then M (A) M(B)
Theorem 2.3.2
Let A be any non-empty subset of a linear space L and α be any scalar, then
M (α A) = α M (A)
Theorem 2.3.3
If A and B are non empty subset of a linear space L, then
M (A B) M (A) + M(B)
2.4. CIRCLED SET
Definition
A set A is said to be circled set if and only if a A A Whenever I A I ≤ 1.
Theorem 2.4.1
If A is a circled set in a real linear space L, then M(A) is circled.
Theorem 2.4_2
If A is a M-Cone then A = M (A)
2.5 W(A), THE SMALLEST CONVEX CIRCLED SET
CONTAINING A:
Theorem 2.5.1
Let P be a linear space and A be any non-empty subset of P, then
M ( A) W (A) M ( A)
[ a] = ½ [a ]=1
2.6. SOME OTHER RESULTS ON A M-CONE:
Theorem 2.6.1
Let A be a circled M — cone in in a real linear pace L. Then [A], the smallest linear subspace containing A is given by
[A] = nA
Theorem 2.6.2
Let A be a M-cone in a linear space L, then
cic
P (A) = U n A
n= 1
Theorem 2.6.3
If A and B are any two sets in a linear space L, such that, 0 A B, then
M (A + B) M (A) + M (B)
Theorem 2.6.4
Let E be a M-cone in a linear space L and x be an element of L. Then
X + λw is a convex, where λ is any scalar.
Theorem 2.6.5
Let A be a non empty subset of a linear space L
Let A be a M-cone and x1x2 be two points in L such that
X1 – x2 ( t2 – t1 ) A wher t1, t2 are real numbers, such that 0 ≤ t1 ≤ t2
Then ,
x1 + t1A x2 + t2A
Theorem 2.6.6
Let E and F be the two linear spaces and let f be a linear mapping from E to F.
Let A E, then,
M (f (A)) = f(M ( A )
2.7. M-CONE IN A TOPOLOGICAL VECTOR SPACE
Suppose T is a topology on a vector space X, such that,
(a) Every point of X is a closed set and
(b) The vector space operations are continuous with respect to T, Then
T is said to be vector topology on X and is called a Topological
Vector Space (these conditions also imply that , T is a Hausdroff
Topology). The closure E of E G X is the intersection of all closed
sets of E.
Theorem 2.7.1
Let X be a topological vector space let A X and A be a M-cone, then A is also a M-cone
Theorem 2.7.2
Let X be a topological vector space in which {0} is an open set. Let A be a M-cone in X. Then A is also closed.
Theorem 2.7.3
Let X be a topological vector space, Let A X and A be a M-cone , let A° ≠ ϕ, Then
A° is also a M-cone.
2.8. CONVEX HULL OF A SET A IN VECTOR SPACE:
The convex hull of A in a vector space L, in short denoted by C(A) is the set of all convex combinations of member of A, that is the set of all sums.
t1x1 + t2x2 + ............... + tnxn
In which xi A, ti ≥ 0, ∑ti = 1, n is arbitrary. Also C(A) is convex and it is the intersection of all convex set containing A.
Theorem 2.8.1
Let A be a subset of a vector space L
Then, C(A) C (M (A))
Theorem 2.8.2
Let Abe a subset of a linear space L, then
B(M(A)) M(B(A))
2.9. PRODUCT OF A LINEAR SPACE
Let L be a linear space and A L and B L, then the set A x B called the product of A and B is defined as
AxB={(a,b)| a A b B }
where (a, b) is the ordered pair of elements "a" and "b"
We want to make L x L into a linear space. For this purpose we define addition and scalar multiplication as follows :
(a , b) + (c d) = ( a + c, b + d) ......................(2.9.1)
a (a b) = (a a , a b) .....................................(2.9.2)
Theorem 2.9.1
Let L be a linear space and A, B be M-cone in L, then A x B is M-cone in L x L, A linear space with the operations as defined in (2.9.1I) and ( .9.111)
Theorem 2.9.2
Let A and B be M-cone in L. Let R AxB and R be a M-cone. Then
projection of R to A and B are also M-cones.
Theorem 2.9.3
Let L be a linear space and A L, B L, then
M (AxB)V M (A) x M (B
Is an element of M (B))
2.10 M-CONE IN A NORMED LINEAR SPACE
A normed linear space is a linear space N in which to each vector x there
corresponds a real number, denoted by || x || and called the norm of x in, in such a mariner that
(1) || X || ≥ 0 and || X || = 0 x = 0
(2) ||X + y || ≤ || x || + || y ||
(3) || ax || = | α | || x ||
Where, is a scalar
Theorem 2.10.1
Let N he a normed linear space and A N, then
M ( ) M A
In case M ( ) is closed
M ( ) =
2.10.2 M-Core of a Sequence
Theorem 2.10.3
If A is sequence in the complex plane C, then
Theorem 2.10.4
Definition the Limit point of a set A is known as the derived set of A , let us denote it by D.
Let A sequence in C and D be its derived set, then
M (D) MC (A)
Theorem 2.10.5
If two sequences have the same M-core, then the M-core of one sequence contains the derived set of the other.
Theorem 2.10.6
If a real sequence A has infinitely many distinct points converges to 0, then
M c (A) = {0}
By theorem (2.3.11)
Mc (B) = α Mc(A)
MC (αA) = α Mc(A)
Theorem 2.10.8
M-core of, sequence is contained in the M-linear hull of its core.
2.11 M-CONICAL FUNCTIONAL
Let L be a linear space on which a real-valued function p is defined, then p is said to be non-negatively homogeneous if
p (ax)= ap (x), whenever α ≥ 0, x D
Theorem 2.11.1
All M-conical functionals from a Cone if the real linear space of all real valued functions defined on a linear space L.
Theorem 2.11.2
Let f, g be M-conical functional such that
f : L→R, where Lis a linear space
and g :R→R
be increasing, that is, if x,y are elements of R such that x ≤ y, then g(x) ≤ g(y), then g of the composition of g with defined by
(g o f) (x) = g (f(x))
is a M-conical functional
Theorem 2.11.3
Let L be the linear space, Let f, g be M-conical functional defined on L, then the function h defined by
h(x) = max (f x), g(x)), x L is also a M-conical functional.
Theorem 2.11.4
Let L be a linear space and p be a M-conical functional defined on L-Let a set A defined by
A = {x L : p (x) ≤ 0}
Then A is a M-cone.
Theorem 2.11.5
f is O convex if f is a M-conical functional.
2.12 ASCENDING CIIAIN CONDITION & MAXIMUM CONDITION
A linear space L is said to satisfy the ascending chain condition (ACC) on M-cones, if whenever
A1 A2 A3 ...............,Aj a M-cone, there
Theorem 2.12.1
A linear space L satisfies the scending chain condition on M-cones if any only if it satisfies the maximum condition of M-cones.
2.13 THE SET B:A AND DJ(A) :
Definition
Let L be a linear space over the field F of scalars. Let A L, B L, A ≠ ϕ, B ≠ ϕ
Let us define a set G in F by
G = {a F:aA B}
We define this set G by B:A
Theorem 2.13.1
Let L be a linear space over F, and A, B be M-cones in L. Then B:A is a M-cone in F.
Definition
Let L be linear space over F of dimension n ≥1, then we confined n elements el, e2,........., en in
L such that if x L, then it can be expressed as
X = λ1e1 + λ2e2 + ......+ λnenλ1’s being scale.
Theorem 2.13.2
Let L be a n-dimensional linear space and A be a M-cone in L. Then Dj(A) is a M-cone for
j = 1,2, ..............,n
Theorem 2.13.3
Let A and B be subset of a linear space L such that A B.
Then Dj (A) Dj(B)
Hence, Dj (A B) Dj(A) Dj(B)
Theorem 2.13.4
Let A and B be subsets of a linear space L. Then
Dj (A B) Dj (A) Dj (B)
3.1 BOUNDED RECTANGUALR SUBSET OF PRODUCT;
We prove some results about the relative distribution of the closure operator in the real compactification of a product of two spaces when one factor is a topological group.
We combine different techniques here when dealing with a bounded subset of the product of two topological spaces.
Theorem 3.2
Let A be a bounded subset of a topological group G and B be the bounded subset of a space Y. If f C* (GxY). L A. and f* is a continuous extension of f over (A*xB*)U(AxB*) then the functions
F*L(q) = inf {f* (x.q) : X L}
And G*L(q) = sup {f* (x.q) : x L} are continuous on B*
Theorem 3.3
Let A and B be bounded subsets of a topological group G and a space Y respectively. Then following relative distribution law is valid.
ClU(Gxy) (AxB) = (AxB)* = A* x B* clUGA
A RING OF OPERATORS ON A HILBERT SPACE
4.1 INTRODUCTION
In this chapter, we have constructed a set of operators R on a Hilbert space H in the following way :
R = { Ti: Ti2 = 0, Ti Tj= 0 for all i, j}
We have shown that R is a Ring as well as a Banach Space and on algebra we define R) , a subset of R as
R= { Ti: Ti = Ti* Ti R}
We have shown that RI is a closed ideal of R.
4.2 A RING OF OPERATORS ON A HIL BERT SPACE:
Let H be a Hilbert Space. Define a set R of operators on H by
R = { Ti: Ti2 = 0, Ti Tj= 0 for all i, j}
Since 02 = 0 and 0. Tj = 0 for all j, it follows that the zero operator belongs to R and R is non-empty. We define addition and multiplication of operators on R as usual.
Theorem 4.2.1
(R, +, . ) is a ring.
Theorem 4.2.2
R is a Banach Space.
Theorem 4.2.3
Let Ri consista of those elements of R which are self adjoins.
Ri= { Ti= Ti = Ti* Ti R}
Then RI is an ideal of R.
Theorem 4.2.4
RI is a closed set.
Theorem 4.2.5
R is an algebra
Thesis Keywords/Search Tags:
Mathematics
This Thesis Abstract may be cited as follows:
No user preference. Please use the standard reference methodology.
| Thesis Images: | |
|
Purushothaman .s (click to enlarge) |
Submission Details: Thesis Abstract submitted by Purushothaman Subramani from India on 07-Mar-2014 12:53.
Abstract has been viewed 1717 times (since 7 Mar 2010).
Purushothaman Subramani Contact Details: Email: purushi.ind@gmail.com Phone: 08685000207
Disclaimer
Great care has been taken to ensure that this information is correct, however ThesisAbstracts.com cannot accept responsibility for the contents of this Thesis abstract titled "Some Of Study On Cone And Topological Space". This abstract has been submitted by Purushothaman Subramani on 07-Mar-2014 12:53. You may report a problem using the contact form.
© Copyright 2003 - 2024 of ThesisAbstracts.com and respective owners.