1. ADEKUNLE AJASIN UNIVERSITY AKUNGBA AKOKO
A PROJECT WORK ON
MAP PROJECTIONS AND THEIR
PROPERTIES
BY
FALADE JOHN TEMIDAYO
MATRIC NO: 110416008
SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND INDUSTRIAL
MATHEMATICS, FACULTY OF SCIENCE, ADEKUNLE AJASIN UNIVERSITY
AKUNGBA AKOKO, ONDO STATE.
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
AWARD OF BACHELOR OF SCIENCE (B.Sc. Hons) DEGREE IN
MATHEMATICS.
NOVEMBER 2015

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CERTIFICATION
This is to certify that this project work was carried out by FALADE JOHN
TEMIDAYO with matriculation number 110416008 in the Department of
Mathematical Science.
_________________________ _______________
Dr. O.S Olusa DATE
Supervisor
_________________________ _______________
Dr. E.P Akpan DATE
Head of Department
_________________________ _______________
External Examiner DATE

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DEDICATION
This project is dedicated to Almighty God.

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ACKNOWLEDGEMENT
My sincere appreciation goes to my project supervisor Dr. O.S Olusa for his
fatherly love, advice, suggestions, his direction, and contribution towards the
completion of this project, may the Lord bless and keep you sir. Special thanks to
the Head of Department of Mathematical Sciences Dr. E.P Akpan and all the
Lecturers in the Department of Mathematical Sciences.
I am very grateful to the contributions and prayers of my parent Mr and Mrs Falade
and that of my guardian Mrs Yusuf Funke. I will not forget the love of my grandpa
Late Chief Adeyemi Falade, may his gentle soul rest in perfect peace. I cannot
forget the encouragement I always receive from my only brother and friend Falade
Olagoke. I also appreciate my only sister Falade Faith and all my siblings for their
word of encouragement. I will never forget the support of my little brother
Aremora Damilare Samuel for been there for me all the time.
I also want to say thank you to Mr. Oni Sunday, Mr. Oluwadare Seyi, Mr. Alafe
Taiwo, Miss. Biola, Sis Tito and all 400 Level 2014/2015 set of Mathematical
Sciences for their love and encouragement.
I will love to show my sincere appreciation to all the Pastors in Deeper Life Bible
Church Oka Region and the Regional Coordinator (Pastor Akinniyi Idowu),
Associate Coordinators (Pastor Olusola Ajayi, Pastor Saliu John) and their wives

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all the Executives of Deeper Life Campus Fellowship (Bro Sunday, Sis Ojumola
Joy, Bro Asade Rotimi, Bro Adewole Jeremiah, Bro Ayodeji Samuel, Sis
Ogedengbe Damilola, Sis Fatoba Kitan, Sis Adebayo Iyanuloluwa, Sis Adebayo
Adesola and others), to all the workers in DLCF AAUA and all the members of the
Fellowship.
Finally, I want to appreciate the effort of the following people: Sis Ayegbo Ruth,
Bro Ajobiewe Jude, Bro Towolawi Olorunwa, Bro Otaraki David, Bro Ayeyemi
Tominiyi, Bro Ogunbodede Deji, Bro Jacob Bola, Sis Olaoye Blessing, Sis
Ologunagba Grace, Bro Bagun Festus, Sis Olanipekun Esther, Bro Oludaisi
Gabriel, Bro Onalo Naphtali, Bro Babayemi Tosin, my Lenovo PC, Sis Adebayo
Sefunmi, Sis Faith Essien, Sis Omoniregun Doyin, Sis Oripelaye Shola, Bro
Olaniyan Kunle and Bro Akande Akinsanmi.

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ABSTRACT
This project explains the concept of Map Projections, their properties and its
application to various mathematical structures.
The relevant definitions, properties, theorems and proofs are given.

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TABLE OF CONTENT
Title Page i
Certification ii
Dedication iii
Acknowledgement iv
Abstract vi
Table of Content vii
CHAPTER ONE INTRODUCTION
1.0 Introduction 1
1.1 Literature Review 4
1.2 Definition of Terms 7
CHAPTER TWO PROPERTIES
2.0 Properties of Map Projections 11
2.1 Introduction 11
2.2 Properties of Metric Projection 11
2.3 Homomorphisms 13
2.4 Projection on Banach Spaces 14

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2.5 Orthogonal Projection 15
2.6 Continuity of the Projection Map 17
2.7 Projection on Topological Group 18
2.8 Generalized and Hyper-Generalized Projection 19
2.9 Projection in Vector Space 23
2.10 Productof Projection 24
2.11 Sum of Projection 25
CHAPTER THREE APPLICATION
3.1 Topological Obstructions and Polytope Projection 27
3.2 Stereographic Projection 30
3.3 Orthogonal Projection 32
3.4 Matrix Projection 33
3.5 Application of Map Projection to the World 37
CHAPTER FOUR CONCLUSION
Conclusion 40
References 41

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CHAPTER ONE
1.0 INTRODUCTION
In mathematics, a projection is a mapping of set (or any mathematical
structures) into a subset or (substructures), which is equal to its square for mapping
composition (or its Idempotent).
The restriction to a subspace of a projection is also called a projection, even if the
idempotent property is lost.
The concept of projection in mathematics is a very old one, most likely
having its roots in the phenomenon of shadows cast by the real world objects to the
ground. This rudimentary idea was refined and abstracted, first in a geometric
context and later in other branches of mathematics. Over time differing version of
the concept developed, but today, in a sufficiently abstract setting, we can unify
these variations.
An everyday example of projection is the casting of shadows (or mapping)
onto a plane. The projection of a point is its shadow on the plane. The shadow of a
point on the plane is this point itself (idempotence).
The shadow of a three dimensional sphere is a circle. Originally, the notion
of projection mapping was introduced in Euclidean geometry to denote the
projection mapping of Euclidean spaces of three dimensional onto a plane in it.
The two main projection mapping of this kind are:
(a) The projection from a point onto a plane or central projection: If C is the
point, called the centre of projection; the projection of a point P different
from C is the intersection with the plane of the line CP. The point C and the

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point P such that line CP is parallel to the plane do not have any image by
the projection.
(b)The projection parallel to a direction D, onto a plane: The image of a point P
is the intersection with the plane of the line parallel to D passing through P.
In an abstract setting, we can generally say that a projection maps is a mapping
of a set or mathematical structures which is idempotent, which means that a
projection is equal to its composition with itself. A projection may also refer to a
mapping that has a left inverse.
Both notions are strongly related as follows: Let P be an idempotent map from a
set E into itself (i.e. 𝑃2
= 𝑃) and 𝐹 = 𝑃(𝐸) be the image of P. If we denote π the
map P viewed as a map from E onto F and by ί the injection of F into E, then we
have𝜋ί = 𝑖𝑑 𝐹.
Conversely, 𝜋ί = 𝑖𝑑 𝐹 means that 𝜋ί is idempotent. Also, if X1 and X2 are non-
void (i.e. non empty) sets, we define projection map
π1: X1 x X2 → X1 and π2: X1 x X2 → X2 by πί(x1, x2) = xί.
The aim of this work is to study map projections in Mathematics, the objectives
consist in studying the properties of this map projection as well as their
application. In this work we will be looking at various properties of map projection
has it has been extended or generalized to various mathematical situations,
frequently but not always related to geometry. Some of the areas in which map
projection has been used with its properties are in Set theory, Canonical
projections, Category theory, Orthogonal projection (Linear algebra), in topology
and other area with properties which will be discussed explicitly in the second
chapter of this work.

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In the third chapter of this work, we will be discussing the application of
projection mapping I different aspect, for example
(a) In the world: projection mapping has enable to map part of the surface of
the Earth onto a plane, i.e. the three dimensional projection
(Cartography)
(b)In set theory: An operation typified by the ίth projection map written projj,
that takes an element x=(x1, x2, x3, . . . , xj, . . . , xk) of a Cartesian product
X1 x X2 x . . . x Xj x . . . x Xk to the value projj(x)=xj.
A mapping that takes an element to its equivalence class under a given
equivalence relation is known as the canonical projection. Also, evaluation map
sends a function f to the value f(x) for a fixed x. the space of functions Yx can be
identified with the Cartesian product 𝜋𝑌ί and the evaluation map is the projection
map from the Cartesian product.
There are also many applications of map projections in Linear algebra that will
be discussed in the third chapter of this work, and other area of application.

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1.1 LITERATURE REVIEW
1.1.1 Definition:
SupposeV = U ⊕W. Define P: V → V as follows. For each v ∈ V, write v =
u + w with u ∈ U and w ∈ W. Then u and w are uniquely defined for each v ∈ V.
Put P(v) = u. It is straightforward to verify the following properties of P.
(i) P ∈ L(V ).
(ii) P2 = P.
(iii) U = Im(P).
(iv) W = null(P).
(v) U and W are both P-invariant.
This linear map P is called the projection onto U along W (or parallel to W) and is
often denoted by P = PU,W.
1.1.2 Definition:
Let V be a vector space. A map 𝐴: 𝑉 → 𝑉 is a projection operator if it is linear
and satisfies
𝐴2
= 𝐴
We shall assume that 𝐴 ∶ 𝑉 → 𝑉 is a projection operator.
Observe that I −A is also then a projection operator: (𝐼 − 𝐴)2
= 𝐼 − 𝐴 (Since
𝐴2
= 𝐴)
If a point y lies in the image of A then it is of the form 𝐴𝑥, for some 𝑥 ∈ 𝑉, and so
then 𝐴𝑦 = 𝐴 (𝐴𝑥) = 𝐴2
𝑥

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𝐴2
𝑥 = 𝐴𝑥 = 𝑦. Thus,
𝐴𝑦 = 𝑦 If and only if y is in the image of A,
Put another way,
Im(𝐴) = ker(I − 𝐴)
Applying this result to the projection operator I − 𝐴 gives
ker(𝐴) = Im(I − 𝐴)
Any vector 𝑥 ∈ 𝑉 can be expressed as
𝑥 = 𝐴𝑥 + (I − 𝐴)𝑥
Where the first term 𝐴𝑥 is clearly in the image of 𝐴 while the second term is
in ker(𝐴). Furthermore, this decomposition is unique since any element y which is
in both ker(A) and Im(𝐴) must be 0 because 𝑦 ∈ Im(𝐴) implies 𝑦 = 𝐴𝑦 while
𝑦 ∈ ker(𝐴) means Ay = 0. [15]
Thus V splits into a direct sum of the subspace Im(𝐴) 𝑎𝑛𝑑 ker(𝐴):
𝑉 = Im(𝐴) ⊕ ker(𝐴)

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1.1.3 Definition
Rendering a picture of a three-dimensional object on a flat computer screen
requires projecting points in 3-Space to points in 2-Space. We discuss only one of
many points in ℜ3
to points in ℜ2
that preserves the natural appearance of an
object.
Parallel projection stimulates the shadow that is cast onto a flat surface by a
far away light source, such as the sun.
Also in Introduction to Linear Algebra with application [8], Jim Defranza
and Daniel Gagliardi in 2009, discussed map projection properties such as
homomorphisms, isomorphisms, epimorphisms and other properties.
Colin Adams and Robert Franzosa in their book [1]discussed about the map
projections on topology.
Looking at some properties of map projections, Erwin Kreyszig in
Introduction to Functional Analysis with Application, made mention of it [10].
A Treatise on projection map written by Thomas Craig [6], 1882, from
University of Michigan, also explained the application of map projections to the
world.

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1.2 DEFINITION OF TERMS
1.2.1 Definition:
Let A and B be a non-empty sets. A relation f from A and B is called a
mapping (or a map or a function) from A to B if for each element x in A there is
exactly one element y in B (called the image of x under f) such that x is a relation f
to y. If f is a mapping from A to B, we write
f: A→B or
𝐴
𝑓
→ 𝐵
1.2.2 Definition
i. Let 𝑓 ∶ 𝐴 → 𝐵. The sets A and B are called, respectively, the domain and co
domain of the mapping f.
ii. A mapping f: X→X such that f(x) =x for all x ɛ X is called the Identity
mapping on X and is denoted by ίx.
iii. Suppose y0 ∈ Y. Define a Constant mapping 𝑓: 𝑋 → 𝑌 by 𝑓(𝑥) = 𝑦0 for
all 𝑥 ∈ 𝑋.
iv. Let A be a non-empty subset of a set X. then the mapping 𝑓: 𝐴 → 𝑋 such
that 𝑓(𝑎) = 𝑎 for each 𝑎 ɛ 𝐴 is called an Inclusion map of A into X.
v. Let f: A → B. The subset of B containing of every element that is the image
of some element in A is called the image (or range) of the mapping f and is
denoted by Imf. That is; 𝑰𝒎𝒇 = {𝑦 ɛ 𝐵 | 𝑦 = 𝑓(𝑥) 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑥 ɛ 𝐴}

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1.2.3 Definition:
Let X and S be sets. The set of all possible mappings from X to S is denoted
by 𝑆 𝑋
. That is; 𝑆 𝑋
= {𝑓 | 𝑓: 𝑋 → 𝑆}
Suppose X and S are finite sets having m and n elements, respectively. Consider
any mapping 𝑓: 𝑋 → 𝑆. The image of any given element in X can be any one
of the n elements in S. Therefore, the m elements in X can be assigned images
in n 𝑥 n 𝑥 n 𝑥 . . . 𝑥 n = nm
ways. This just means that there are exactly nm
distinct mapping from X to S. Thus, for finite sets X and S, we have the result
that |Sx|=|S||x|
1.2.4 Example
Let X = {0, 1}. Then there are 22=4 distinct mapping from X to X
f1: X → X with 0 0, 1 0
f2: X → X with 0 0, 1 1
f3: X → X with 0 1, 1 0
f4: X → X with 0 1, 1 1
1.2.5 Definitions
A mapping 𝑓: 𝐴 → 𝐵 is
a) Injective (or one-to-one) if, for all 𝑥1, 𝑥2ɛ 𝐴, 𝑥1 ≠ 𝑥2 𝑓(𝑥1) ≠ 𝑓(𝑥2) i.e. if
x1 and x2 are distinct elements of A, then f(x1) and f(x2) are distinct elements
of B.
b) Surjective (or onto) if, for every y ɛ B, y = f(x) for some x ɛ A. That is the
image is the range i.e. if y ɛ B, f -1(y) is a non-empty subset of X

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c) Bijective (or one-to-one-correspondence) if f is injective and surjective. In
this case, there is a function f-1:B → A with f-1 ° f = IA: A → A and f ° f -1 =
IB: B → B. Note that f -1: B → A is also bijective and (f -1)-1 = f.
d) Let f : A → B. A mapping 𝑔: 𝐵 → 𝐴 is called an inverse of f if f ° g =IB and
g ° f = IA.
e) A mapping g: B → A is
i. A left inverse of f if g ° f = IA.
ii. A right inverse of f if f ° g =IB
f) If 𝑓: 𝐴 → 𝐵 is a mapping and 𝑆 ∁ 𝐴, the mapping from S to B given by a → f
(a) for a ɛ S is called the restriction of f to S and is denoted by f |S:S →B
1.2.6 Examples of Induced Mappings
a) Let f: A → B. Let S Ϲ A and T Ϲ B such that f(x) ɛ T for all x ɛ S. Then f
induces the mapping g: S → T given by g(x) = f(x) for all x ɛ S. The mapping
g is also called the restriction of f.
b) There is an important mapping determined by a subset of a set. Let S Ϲ X
and let A = {0, 1}. Then S determines the mapping fS: X → A given by
𝑓(𝑥) = 1 𝑖𝑓 𝑥 ɛ 𝑆
𝑓(𝑥) = 1 𝑖𝑓 𝑥 ɛ 𝑆
The mapping 𝑓𝑆 is called the characteristics function of S.
Conversely, given a mapping 𝑓: 𝑋 → 𝐴, let S = {𝑥𝜀𝑋 | 𝑓(𝑥) = 1}
Then it is clear that the characteristic function of S is the given mapping f.
Moreover, S is the unique subset of X with f as its characteristic function.
This proves that the mapping 𝛹: Ƥ(𝑥) → 𝐴 𝑥
by 𝑆 → 𝑓𝑠 , is bijective.

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It is clear that we could have the set 2 = {1, 2} in a place of A. Thus, there is
a one-to-one-correspondence between the set Ƥ(x) and 2 𝑋
. Because of this,
the set Ƥ(x) is also written as 2 𝑋
.
c) Let E be an equivalence relation on the set X. Then E induces the surjective
mapping Ƥ: X → X|E, with x → E(x) is the equivalence class of x under E.
The mapping is called the canonical mapping from X to the quotient set
X|E.
d) Given sets S and T, there are two canonical mapping with domain S x T –
namely, p: S x T → S with (x, y) → x for all (x, y) ɛ S x T and
q: S x T → with (x, y) → y for all (x, y) ɛ S x T.
The mapping p and q are called the projections from S x T onto S and T
respectively.

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CHAPTER TWO
PROPERTIES OF MAP PROJECTIONS
2.1 INTRODUCTION
Here in this chapter, we shall be looking at the properties of map projections in
many areas. It follows that projection mapping play a key role in many areas. In
particular we shall be discussing the following; Properties of metric projection,
projection in Banach space, properties of projections on topological spaces,
generalized and hyper-generalized projection mapping on Hilbert space,
Orthogonal projections, Isomorphism, Homomorphism, Continuity of projection
mappings, and the Product and Sum of projection.
2.2 PROPERTIES OF METRIC PROJECTION
2.2.1 Definition: Let D Ϲ H be a nonempty subset and let x ɛ H. The point
y ɛ D is called the metric projection of a point x onto a subset D, if for any
z ɛ D there holds the inequality.
||𝑦 – 𝑥|| ≤ ||𝑥 − 𝑦||
The metric projection of a point x onto D is denoted by 𝑃𝐷 𝑥.
2.2.2Theorem: [2] Let 𝑥𝜀𝐻, 𝐷∁𝐻 be a nonempty, con vex and closed
subset and let 𝑦𝜀𝐷. The following conditions are equivalent:
(i) y = PD x,
(ii) < 𝑥 – 𝑦, 𝑧 – 𝑦 > ≤ 0 for all 𝑧 ɛ 𝐷.

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Proof:
(i) (ii). Let y = PD (x), z ɛ D and let 𝑧 = 𝑦 + 𝜆(𝑧 − 𝑦) for 𝜆 ɛ (0; 1).
Obviously, 𝑧𝜆 𝜀𝐷 since D is convex. We have by the properties of the scalar
product
|| 𝑥 – 𝑦||2
≤ || 𝑥 − 𝑧𝜆||2
= || 𝑥 – 𝑦 – 𝜆(𝑧 – 𝑦) ||2
= || 𝑥 – 𝑦||2
– 2𝜆‹𝑥 – 𝑦, 𝑧 – 𝑦›+ 𝜆2
|| 𝑧 – 𝑦 ||2
Since ƛ> 0, we have
< 𝑥 – 𝑦, 𝑧 – 𝑦 > ≤ 𝜆
2⁄ || 𝑧 − 𝑦||2
If we let 𝜆 → 0 in the last inequality, we obtain (ii) the limit.
(ii) (i). By the properties of the scalar product and by (ii) we obtain
for any z ɛ D
||𝑧 − 𝑥||2
= ||𝑧 − 𝑦 + 𝑦 − 𝑥||2
= ||𝑧 − 𝑦||2
+ ||𝑦 − 𝑥||2
+ 2‹𝑧 – 𝑦, 𝑦 – 𝑥›
≥ ||𝑦 − 𝑥||2
,
which, by the definition of the metric projection, gives (ii)
2.2.3 Corollary:Let 𝐷 Ϲ 𝐻 be nonempty, convex and closed. Then Fix PD= D.
Consequently, the metric projection PD is an idempotent operator.
Proof:
If x ɛ D, then it follows from the definition of the metric projection that
x = PD x. If x ɛ D, then x = PD x since PD x.

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2.3 HOMOMORPHISMS
2.3.1 Definition: [7] If G and G’ are multiplicative groups, a function 𝑓 ∶ 𝐺 → 𝐺’
is a homomorphism if, for all 𝑎, 𝑏 ∈ 𝐺, 𝑓(𝑎 · 𝑏) = 𝑓(𝑎) · 𝑓(𝑏).
On the left side, the group operation is in G, while on the right side it is in G’. The
kernel of f is defined by
𝑘𝑒𝑟(𝑓) = 𝑓−1
(𝑒’) = {𝑎 ∈ 𝐺 ∶ 𝑓(𝑎) = 𝑒’}.
In other words, the kernel is the set of solutions to the equation f(x) =e’. (If G’ is
an additive group, 𝑘𝑒𝑟(𝑓) = 𝑓−1
(0’). Examples The constant map f: G → G’
defined by 𝑓(𝑎) = 𝑒’ is a homomorphism. If H is a subgroup of G, the inclusion
i: H ↪ G is a homomorphism. The function 𝑓: 𝑍 → 𝑍 defined by 𝑓(𝑡) = 2𝑡 is a
homomorphism of additive groups, while the function defined by 𝑓(𝑡) = 𝑡 + 2 is
not a homomorphism. The function ℎ: 𝑍 → ℝ − {0} defined by ℎ(𝑡) = 𝑡2
is a
homomorphism from an additive group to a multiplicative group.
2.3.2 Theorem: Suppose G1 and G2 are additive groups. Define an addition on
G1×G2 by (a1,a2)+(b1,b2)=( a1 +b1,a2 +b2). This operation makes G1×G2 into a
group. Its “zero” is (01,02) and −(a1,a2)=(−a1,−a2). Theprojections π1: G1 ×G2 →
G2 and π2: G1 ×G2 → G2 are group homomorphisms. Suppose G is an additive
group. We know there is a bijection from {functions f: G → G1 ×G2} to {ordered
pairs of functions (f1,f2) where f1 : G → G1 and f2 : G → G2}. Underthis bijection, f
is a group homomorphism iff each of f1 and f2 is a group homomorphism.

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Proof:
It is transparent that the product of groups is a group, so let’s prove the last part.
Suppose G, G1, and G2 are groups and f = (f1,f2) is a function from G to G1 × G2.
Now f(a + b)=( f1(a + b),f2(a + b))
And f(a)+f(b)= (f1(a),f2(a))+(f1(b),f2(b)) = (f1(a)+f1(b),f2(a)+f2(b)).
An examination of these two equations shows that f is a group homomorphism if
and only if each of f1 and f2 is a group homomorphism.
2.3.3 Definition: If f: R → R’ is a bijection which is a ring homomorphism, then
f-1: R’ → R is a ring homomorphism. Such an f is called a ring isomorphism. In the
case R = R’, f is also called a ring automorphism.
2.4 PROJECTION ON BANACH SPACES
A projection in a Banach space is a continuous linear mapping P of the space into
itself which is such that P2=P. Two closed linear manifolds M and N of a Banach
space B are said to be complementary if each z of B is uniquely representable as
x+y, where x is in M and y in N.
This is equivalent to the existence of a projection for which M and N are the range
and null space. It is therefore also true that closed linear subsets M and N of B are
complementary if and only if the linear span of M and N is dense in B and there is
a number 𝜀 > 0 such that ||x + y|| ≥ ε||x|| if x is in M and y in N. It is known
that a Banach space M is complemented in each Banach space in which it can be
embedded if it is isomorphic with a complemented subspace of the space (m) of
bounded sequences. In particular, if M is a subspace of a Banach space Z and is

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isometric with a subspace M' of (m), then there is a projection of Z onto M of norm
less than or equal to X if there is a projection of (m) onto M' of norm equal to ƛ.
Thus the existence of a complement in (m) for a subspace M of (m) is independent
of the method by which M is embedded in (m). Any separable Banach space is
isometric with a subspace of (m). Hence a separable Banach space is
complemented in each space in which it can be embedded if and only if it is
complemented in (m). It has been conjectured that no separable subspace of (m) is
complemented in (in), or perhaps that separable complemented subspaces of (m)
are reflexive. There are two classes of separable Banach spaces which are known
to be not complemented in (m). These are the separable Banach spaces which have
a subspace isomorphic with (c0) and the separable Banach spaces whose first
conjugate space is not weakly complete, of which h is an example. It is interesting
to note that (c0) is complemented in any separable Banach space B in which it can
be embedded. In particular, the projection of B onto (c0) can be of norm 2 but there
are separable Banach spaces containing (c0) for which the norm of a projection
onto (c0) must have norm as large as 2.
2.5 ORTHOGONAL PROJECTION
We begin by describing some algebraic properties of projections. If M and N are
subspaces of a linear space X such that every x ∈ X can be written uniquely as
x = y+z with y ∈ M and z ∈ N,
then we say that X = M⊕N is the direct sum of M and N, and we call N a
complementary subspace of M in X.
The decomposition x = y + z with y ∈ M and z ∈ N is unique if and only if

24. 24
M ∩N = {0}. A given subspace M has many complementary subspaces. For
example, if X = R3 and M is a plane through the origin, then any line through the
origin that does not lie in M is a complementary subspace. Every complementary
subspace of M has the same dimension, and the dimension of a complementary
subspace is called the co-dimension of M in X.
If X = M⊕N, then we define the projection
P: X → X of X onto M along N by Px = y,
where x = y +z with y ∈ M and z ∈ N. This projection is linear, with
ranP = M and kerP = N,
and satisfies P2 = P. As we will show, this property characterizes projections, so we
make the following definition.
2.5.1 Definition: A projection on a linear space X is a linear map
𝑃: 𝑋 → 𝑋 such that
𝑃2
= 𝑃
Any projection is associated with a direct sum decomposition.
2.5.2 Theorem: [11] Let X be a linear space.
(a) If P: X → X is a projection, then X = ranP ⊕kerP.
(b) If X = M ⊕N, where M and N are linear subpaces of X, then there is a
projection P: X → X with ranP = M and kerP = N.
Proof.
To prove (a), we first show that x ∈ ranP if and only if x = Px.

25. 25
If x = Px, then clearly x ∈ ranP. If x ∈ ranP, then x = Py for some y ∈ X, and since
P2= P, it follows that Px = P2 y = Py = x.
If 𝑥 ∈ 𝑟𝑎𝑛𝑃 ∩ 𝑘𝑒𝑟𝑃 then 𝑥 = 𝑃𝑥 and 𝑃𝑥 = 0, so 𝑟𝑎𝑛𝑃 ∩ 𝑘𝑒𝑟𝑃 = {0}. If x ∈
X, then we have 𝑥 = 𝑃𝑥 + (𝑥 − 𝑃𝑥), where 𝑃𝑥 ∈ 𝑟𝑎𝑛𝑃 𝑎𝑛𝑑 (𝑥 − 𝑃𝑥) ∈
𝑘𝑒𝑟𝑃, since
𝑃 (𝑥 − 𝑃𝑥) = 𝑃𝑥 − 𝑃2
𝑥 = 𝑃𝑥 − 𝑃2
𝑥 = 0.
𝑇ℎ𝑢𝑠 𝑋 = 𝑟𝑎𝑛𝑃 ⊕ 𝑘𝑒𝑟𝑃.
To prove (b), we observe that if
X = M ⊕ N,
then x ∈ X has the unique decompositionx = y + z with y ∈ M and z ∈ N, and Px =
y defines the required projection.
2.6 Continuity of the projection map
The invariance of the distance under translations yields the following
consequence for the projection onto a fixed convex subset.
2.6.1 Theorem: [8] Let C be a nonempty closed convex subset of a uniformly
convex Banach space X.Let γ be a gage of monotonicity of the duality mapping Jh
on the ball rBX , with r > d(0, C). Let q > 0 be such that 3q + d(0, C) < r . Then,
the projection mapping pC is uniformly continuous on qBX with
||pC (w) – pC (w’) || ≤ γ−1 2h(r)||w – w’||+ || w – w’ ||
||I − pC) (w) −I – pC (w’)|| ≤ γ−1 2h(r ) ||w – w’||
Proof:

26. 26
Let t: = 2q + d(0, C), so that q + t< r . Given w,w’ ɛ qBX, let us set
C:= C + w – w’, u: = pC (w’)
Then, observing that, for any y ∈ C,
we have
|| (𝑢 + 𝑤 – 𝑤’) – 𝑤|| = 𝑢 – 𝑤 ≤ ||𝑦 – 𝑤’|| = ||(𝑦 + 𝑤 – 𝑤’)– 𝑤’||
𝑃𝑐( 𝑤) = 𝑝𝑐 ( 𝑤’) + 𝑤 – 𝑤’
.
Since by definition of C we have 𝑑(𝐶, 𝐶’ ) ≤ ||𝑤 – 𝑤’|| , we note that
𝑑(0, 𝐶’ ) ≤ 𝑑(0, 𝐶) + 𝑑(𝐶, 𝐶’ ) ≤ 𝑡
and 𝐶, 𝐶’ ∈ 𝑡(𝑋 ). Thus, we have
Since
||Pc(w)− pC (w’)|| ≤ ||pC (w) – pC (w’) + ||pC (w) – Pc(w’ )
≤ γ-1 2h(r )||w – w’||) + || w – w’||(w) – pCc(w),
we get the second inequality.
2.7 PROJECTION ON TOPOLIGICAL GROUP
2.7.1 Theorem:Let G be a topological group and H be a subgroup of G. Then the
canonical projection 𝓅: 𝐺 → 𝐺 = 𝐻 is an open map.
Proof:
Let V be an open subset of G. Then it implies that the image ƥ (V) is open in G/H if
and only if 𝓅 -1(𝓅 (V)) is open in G. This set however is easily seen to equal VH,

27. 27
which is the union of the translate V h as h ranges through H and is therefore an
open set.
2.7.2 Definition: Let H be a subgroup of a topological group G. The quotient
topology on G/H is defined such that a set U is a subset of G/H and U is open if
and only if 𝓅−1
(U ) is open in the topology of G, where 𝓅 ∶ 𝐺 → 𝐺/𝐻 is the
canonical projection
2.8 GENERALIZED AND HYPER-GENERALIZED PROJECTION
Let H be a separable Hilbert space and L(H ) be a space of all bounded linear
operators on H . The symbols R(A), N (A) and A∗ denote range, null space and
adjoint operator of operator A ∈ L(H ). Operator A ∈ L(H ) is a projection
(idempotent) if 𝐴2
= 𝐴, while it is an orthogonal projection if 𝐴
∗
= 𝐴 = 𝐴2
.
Operator is hermitian (self adjoint) if 𝐴 = 𝐴∗
, normal if 𝐴𝐴∗
= 𝐴∗
𝐴 and unitary if
𝐴𝐴∗
= 𝐴∗
𝐴 = 𝐼 . All these operators have been extensively studied and there are
plenty of characterizations both of these operators and their linear combinations.
The Moore-Penrose inverse of 𝐴 ∈ 𝐿(𝐻 ), denoted by 𝐴†
, is the unique solution
to the equations
𝐴𝐴†
𝐴 = 𝐴, 𝐴†
𝐴𝐴†
= 𝐴†,
(𝐴𝐴†
)∗
𝐴𝐴†
,(𝐴†
𝐴)∗
= 𝐴†
𝐴.
Notice that A exists if and only if R(A) is closed. Then 𝐴𝐴†
is the orthogonal
projection onto R(A) parallel to 𝑁 (𝐴∗
), and 𝐴†
𝐴 is the orthogonal projection onto
R(A∗) parallel to N (A).
Consequently, 𝐼 − 𝐴𝐴†
is the orthogonal projection onto N (A∗) and 𝐼 − 𝐴†
𝐴 is the

28. 28
orthogonal projection onto N (A).
For 𝐴 ∈ 𝐿(𝐻 ), an element 𝐵 ∈ 𝐿(𝐻 ) is the Drazin inverse of A, if the
following hold:
𝐵𝐴𝐵 = 𝐵, 𝐵𝐴 = 𝐴𝐵, 𝐴 𝑛+1
𝐵 = 𝐴 𝑛
for some non-negative integer n. The smallest such n is called the Drazin index of
A. By AD we denote Drazin inverse of A and by ind(A) we denote Drazin index of
A.
If such n does not exist, 𝑖𝑛𝑑(𝐴) = ∞ and operator A is generalized Drazin
invertible. Its inverse is denoted by Ad. Operator A is invertible if and only if
𝑖𝑛𝑑(𝐴) = 0.
If 𝑖𝑛𝑑(𝐴) ≤ 1, operator A is group invertible and AD is its group inverse, usually
denoted by 𝐴⋕
.
Notice that if the Drazin inverse exists, it is unique. Drazin inverse exists if R(An)
is closed for some non-negative integer n.
Operator A ∈ L(H ) is a partial isometry if AA∗ A = A or, equivalently, if 𝐴†
=
𝐴∗
. Operator A ∈ L(H ) is EP if 𝐴𝐴†
= 𝐴†
𝐴, or, in the other words, if 𝐴†
=
𝐴 𝐷
= 𝐴⋕
. Set of all EP operators on H will be denoted by £Ƥ(H ). Self-adjoint
and normal operators with closed range are important subset of set of all EP
operators. However, converse is not true even in a finite dimensional case.
These operators extend the idea of orthogonal projections by deleting the
idempotency requirement. Namely, we have the following definition:

29. 29
2.8.1 Definition: Operator A ∈ L(H ) is
(a) a generalized projection if 𝐴2
= 𝐴∗
;
(b) a hyper-generalized projection if 𝐴2
= 𝐴†
.
The set of all generalized projection on H is denoted by GP (H) and the set of all
hyper- generalized projection is denoted by HGP (H).
2.8.2 Characterization of generalized and hyper-generalized
projections
2.8.2.1 Theorem: [13] Let A ∈ L(H ). Then the following conditions are
equivalent:
(a) A is a generalized projection.
(b) A is a normal operator and A4 = A.
(c) A is a partial isometry and A4 = A.
Proof: (a ⇒ b) Since
𝐴𝐴∗
= 𝐴𝐴2
= 𝐴3
= 𝐴2
𝐴 = 𝐴∗
𝐴,
𝐴4
(𝐴2
)2
= (𝐴∗
)2
= (𝐴2
)∗
= (𝐴∗
)∗
= 𝐴,
the implication is obvious.
(b ⇒ a) If 𝐴𝐴∗
= 𝐴∗
𝐴, recall that then exists a unique spectral measure E on the
Borel subsets of 𝜎(𝐴) such that 𝐸(∆) is an orthogonal projection for every subset
∆⊂ 𝜎( 𝐴), 𝐸(∅) = 0, 𝐸( 𝐻 ) = 𝐼 and if ∆ 𝑖 ∩ ∆ 𝑗 = ∅for 𝑖 ≠ 𝑗, then 𝐸(∪

30. 30
∆ 𝑖 ) = 𝐸(∑∆ 𝑖).
Moreover, A has the following spectral representation
𝐴 = ∫ 𝜆𝑑𝐸𝜆 , 𝜎(𝐴) 𝑤ℎ𝑒𝑟𝑒 𝐸𝜆 = 𝐸(𝜆) is the spectral projection associated
with the point 𝜆 ∈ 𝜎(𝐴).
From 𝐴4
= 𝐴, we conclude 𝐴3
𝑅(𝐴) = 𝐼𝑅(𝐴) and 𝜆3
= 1, or, equivalently
𝜎(𝐴) ⊆ {0,1, 𝑒
2𝜋𝑖
3⁄
, 𝑒
−2𝜋𝑖
3⁄
}. Now,
𝐴 = 0𝐸(0) ⊕ 1𝐸(1) ⊕ 𝑒
2𝜋𝑖
3⁄
𝐸 (𝑒
2𝜋𝑖
3⁄
) ⊕ 𝑒
−2𝜋𝑖
3⁄
𝐸 (𝑒
−2𝜋𝑖
3⁄
), here E(λ) is
the spectral projection of A associated with the point 𝜆 ∈ 𝜎( 𝐴) such that 𝐸( 𝜆) =
0 if 𝜆 ∈ 𝜎( 𝐴), 𝐸( 𝜆) = 0 if 𝜆 ∈ {0, 1, 𝑒
2𝜋𝑖
3⁄
, 𝑒
−2𝜋𝑖
3⁄ }
σ(A)⁄ and 𝐸(0) ⊕
𝐸(1) ⊕ 𝐸 (𝑒
2𝜋𝑖
3⁄
) ⊕ 𝐸(𝑒
−2𝜋𝑖
3⁄
) = 𝐼 . From the fact that 𝜎(𝐴2
) = 𝜎(𝐴∗
) and
from uniqueness of spectral representation, we get 𝐴2
= 𝐴∗
.
(a ⇒ c) If 𝐴∗
= 𝐴2
, then 𝐴4
= 𝐴𝐴2
𝐴 = 𝐴𝐴∗
𝐴 = 𝐴. Multiplying from the left
(from the right) by 𝐴∗
, we get 𝐴∗
𝐴𝐴∗
𝐴 = 𝐴∗
𝐴 (𝐴𝐴∗
𝐴𝐴∗
= 𝐴𝐴∗
), which proves
that 𝐴∗
𝐴 (𝐴𝐴∗
) is the orthogonal projection onto 𝑅(𝐴∗
𝐴)= 𝑅(𝐴∗
) = 𝑁 (𝐴)
⊥
(𝑅(𝐴𝐴∗
) = 𝑅(𝐴) = 𝑁(𝐴∗
)
⊥
), i.e., 𝐴(𝐴∗
) is a partial isometry.
(c ⇒ a) If A is a partial isometry, we know that AA* is orthogonal projection onto
𝑅(𝐴𝐴) = 𝑅(𝐴). Thus, 𝐴𝐴∗
𝐴 = 𝑃 𝑅 ( 𝐴)
𝐴 = 𝐴 and 𝐴 𝐴2
= 𝐴∗
= 𝐴𝐴2
𝐴 = 𝐴
implies 𝐴2
= 𝐴∗

31. 31
2.9 Projection in Vector Space
Let V be a vector space over a field K. Let B={x1, x2, x3,…, xn} be a base of V over
K. Then for all x∈V there exist unique scalars λ1, λ2,…,λn in K such that
∑𝑛𝑖 = 1𝜆𝑖 𝑥𝑖 = 𝑥.
We call the 𝜆𝑖 the i-th component of x with respect to base B. Let pi(x) denote the
𝑖 − 𝑡ℎ component of x with respect to base B, where 𝑝𝑖 is a projection map.
2.9.1 Theorem:
1. 𝑃𝑖: 𝑉 → 𝐾 is a linear map and therefore 𝑝𝑖 ∈ 𝐻𝑜𝑚𝑘(𝑉, 𝐾).
2. {p1,p2,…,pn} is a base of 𝐻𝑜𝑚𝐾(𝑉, 𝐾).
3. There is an isomorphism 𝛹: 𝑉 → 𝐻𝑜𝑚𝐾(𝑉, 𝐾) such that 𝛹(𝑥𝑖) = 𝑝𝑖 for all
1 ≤ 𝑖 ≤ 𝑛.
Proof:
Let 𝑣, 𝑤 ∈ 𝑉, where 𝑣 = 𝑎1 𝑥1 + ⋯ 𝑎 𝑛 𝑥 𝑛 and 𝑤 = 𝑏1 𝑥 𝑛 + ⋯ 𝑏 𝑛 𝑥 𝑛. Since 𝑣 + 𝑤 =
(𝑎1 + 𝑏1)𝑥1 + ⋯+ (𝑎 𝑛 + 𝑏 𝑛)𝑥 𝑛, we have 𝑝𝑖(𝑣 + 𝑤) = 𝑎𝑖 + 𝑏𝑖 = 𝑝𝑖(𝑣) + 𝑝𝑖(𝑤).
Now suppose 𝑐 ∈ 𝐾. Clearly,𝑐𝑣 = (𝑐𝑎1)𝑥1 + ⋯+ (𝑐𝑎 𝑛)𝑥 𝑛, and thus 𝑝𝑖(𝑐𝑣) =
𝑐𝑎𝑖 = 𝑐𝑝𝑖(𝑣).
Assume 𝑐𝑖 ∈ 𝐹 such that
𝑐1 𝑝1 + ⋯+ 𝑐 𝑛 𝑝 𝑛 = 0 ∈ 𝐻𝑜𝑚𝐾(𝑉, 𝐾)
If 𝑐𝑖 ≠ 0 for some i, we have
(c1 p1+⋯+cn pn)(xi)=c1 p1(xi)+⋯+cn pn(xi)=ci pi(xi)=ci∈K

32. 32
in which case c1 p1+⋯+cn pn is not zero. Alternatively, it's not too hard to show that
{p1,…,pn} is a spanning set for 𝐻𝑜𝑚𝐾(𝑉, 𝐾). Any 𝑇 ∈ 𝐻𝑜𝑚𝐾(𝑉, 𝐾) is completely
determined by its value at the basis elements, so say we have T(xi)=λi. Then
(λ1 p1+⋯+λn pn)(xi)=λ1 p1(xi)+⋯+λn pn(xi)=λi pi(xi)=λi
Since both maps agree on basis elements, we conclude that λ1 p1+…+λn pn=T.
2.10 Theorem: [10] (Product of projections). [1]In connection with products
(composites) of projections on a Hilbert space 𝐻, the following two statements
hold.
(a) 𝑃 = 𝑃1 𝑃2 is a projection on 𝐻 if and only if the projections 𝑃1 and 𝑃2 commute,
that is 𝑃1 𝑃2 = 𝑃2 𝑃1 . Then 𝑃 projects 𝐻 onto 𝑌 = 𝑌1 ∩ 𝑌2, where 𝑌𝑗 = 𝑃𝑗(𝐻).
(b) Two closed subspaces 𝑌and 𝑉 of 𝐻 are orthogonal if and only if the
corresponding projections satisfy 𝑃𝑌 𝑃𝑉 = 0.
Proof: (a) Suppose that 𝑃1 𝑃2 = 𝑃2 𝑃1 , then 𝑃 is self-adjoint. 𝑃 is idempotent since
𝑃2
= ( 𝑃1 𝑃2 )( 𝑃1 𝑃2 ) = 𝑃1
2
𝑃2
2
= 𝑃1 𝑃2 = 𝑃
Hence 𝑃 is a projection, and for every 𝑥 ∈ 𝐻 we have
𝑃𝑥 = 𝑃1 ( 𝑃2 𝑥) = 𝑃2( 𝑃1 𝑥)
Since 𝑃1 projects 𝐻 onto 𝑌1, we must have 𝑃1( 𝑃2 𝑥) 𝜖 𝑌1. Similarly, 𝑃2( 𝑃1 𝑥) 𝜖 𝑌2.
Together 𝑃𝑥 ∈ 𝑌1 𝑛 𝑌2,since 𝑥 ∈ 𝐻 was arbitrary, this shows that 𝑃
projects 𝐻 𝑖𝑛𝑡𝑜 𝑌 = 𝑌1 ∩ 𝑌2 . Actually, 𝑃 projects 𝐻 𝑜𝑛𝑡𝑜 𝑌. Indeed, if 𝑦 ∈ 𝑌,
then, 𝑦 ∈ 𝑌1, 𝑦 𝐸 𝑌2 , and
𝑃𝑦 = 𝑃1 𝑃2 = 𝑃1 𝑦 = 𝑦.

33. 33
Conversely, if 𝑃 = 𝑃1 𝑃2 is a projection defined on 𝐻, then 𝑃 is self-adjoint, and
𝑃1 𝑃2 = 𝑃2 𝑃1 .
(b) If 𝑌 ⊥ 𝑉, then 𝑌 ∩ 𝑉 = {0} and 𝑃𝑦 𝑃𝑣 𝑥 = 0 for all 𝑥 ∈ 𝐻 by part (a), so
that 𝑃𝑦 𝑃𝑣 𝑥 = 0. Conversely, if 𝑃𝑦 𝑃𝑣 𝑥 = 0, then for every 𝑦 ∈ 𝑌 and 𝑣 ∈ 𝑉 we
obtain 〈 𝑦, 𝑣〉 = 〈 𝑃𝑦 𝑦𝑃𝑣 𝑣〉 = 〈 𝑦, 𝑃𝑦 𝑃𝑣 𝑣〉 = 〈 𝑦, 0 〉 = 0.
Hence
𝑌 ⊥ 𝑉.
2.11 Theorem (Sum of projections). Let 𝑃1 and 𝑃2 be projections on a Hilbert
space 𝐻. Then:
(a) The sum 𝑃 = 𝑃1 + 𝑃2 is a projection on 𝐻 if and only if 𝑌1 =
𝑃1 (𝐻) 𝑎𝑛𝑑 𝑌2 = 𝑃2(𝐻) are orthogonal.
(b) If 𝑃 = 𝑃1 + 𝑃2 is a projection, P projects 𝐻 onto 𝑌 = 𝑌1⨁ 𝑌2.
Proof: (a) If 𝑃 = 𝑃1 + 𝑃2 is a projection, 𝑃 = 𝑃2
, written
𝑃1 + 𝑃2 = ( 𝑃1 + 𝑃2) = 𝑃1
2
+ 𝑃1 𝑃2 + 𝑃2 𝑃1 + 𝑃2
2
On the right, 𝑃1
2
= 𝑃1 and 𝑃2
2
= 𝑃2. There remains
𝑃1 𝑃2 + 𝑃2 𝑃1 = 0
Multiplying by 𝑃2 from the left, we obtain
𝑃2 𝑃1 + 𝑃2 𝑃1 = 0

34. 34
Multiplying this by 𝑃2 from the right, we have 2𝑃2 𝑃1 𝑃2 = 0, so that 𝑃2 𝑃1 = 0 and
𝑌1 ⊥ 𝑌2.
Conversely, if 𝑌1 ⊥ 𝑌2, then 𝑃1 𝑃2 = 𝑃2 𝑃1 = 0. This implies 𝑃2
= 𝑃. Since 𝑃1
and 𝑃2 are self-adjoint, so is 𝑃 = 𝑃1 + 𝑃2 . Hence 𝑃 is a projection.
(b) We determine the closed subspace 𝑌 ⊂ 𝐻 onto which 𝑃 projects. Since 𝑃 =
𝑃1 + 𝑃2 , for every 𝑥 ∈ 𝐻 we have
𝑦 = 𝑃𝑥 = 𝑃1 𝑥 + 𝑃2 𝑥
Here 𝑃1 𝑥 ∈ 𝑌1 and 𝑃2 𝑥 ∈ 𝑌2. Hence 𝑦 ∈ 𝑌1 ⊕ 𝑌2 , so that 𝑌 ⊂ 𝑌1 ⊕ 𝑌2. We
show that 𝑌 ⊃ 𝑌1 ⊕ 𝑌2 Let 𝑣 ∈ 𝑌1 ⊕ 𝑌2be arbitrary. Then 𝑣 = 𝑦1 + 𝑦2 ·
Here, 𝑦1 ∈ 𝑌1 and 𝑦2 ∈ 𝑌2. Applying P and using 𝑌1 ⊥ 𝑌2, we thus obtain
𝑃𝑣 = 𝑃1 ( 𝑦1 + 𝑦2) + 𝑃2( 𝑦1 + 𝑦2) = 𝑃1 𝑦1 + 𝑃2 𝑦2 = 𝑦1 + 𝑦2 = 𝑣.
Hence, 𝑣 ∈ 𝑌 𝑎𝑛𝑑 𝑌 ⊃ 𝑌1 ⊕ 𝑌2. Together, 𝑌 = 𝑌1 ⊕ 𝑌2.

35. 35
CHAPTER THREE
APPLICATION OF MAP PROJECTION
3.1 TOPOLOGICALOBSTRUCTIONS AND POLYTOPE PROJECTIONS
We devise a criterion for projections of polytopes that allows us to state when a
certain subcomplex may be strictly preserved by a projection. We associate an
embedding problem to the projection problem. Then we describe methods from
combinatorial topology, which may yield obstructions to the associated
embeddability problem. Finally, we specialize the obstructions to the problem of
preserving certain skeleta of polytopes by projections.
3.1.1 Associated Polytope And Subcomplex
We build a bridge between projection problems and embeddability problems as
follows: We associate a polytope with certain simplex faces to a projection of a
polytope with certain strictly preserved faces via Gale duality. The simplex faces
of this associated polytope form a simplicial complex. If we can show that this
simplicial complex cannot be embedded into the boundary of the associated
polytope, then there is no realization of the polytope that allows for a projection
preserving the given subcomplex.
Sanyal uses the same approach to analyze the number of vertices of Minkowski
sums of polytopes, since Minkowski sums are projections of products of polytopes.
The vertices of a (simple) polytope P give rise to a simplicial complex Σ0. If 𝜋 ∶
𝑃 → 𝜋(𝑃) is a projection preserving the vertices, then Σ0 is realized in a
(simplicial) sphere whose dimension depends on dimπ(P). So if the simplicial
complex Σ0 cannot be embedded into that sphere then there exists no realization of
the polytope such that all vertices survive the projection.

36. 36
Proposition 3.1.2: [14] Let π :ℝ 𝑑
→ℝ 𝑒
be the projection to the first e coordinates
of a d-polytope 𝑃 given by its facet inequalities (A(e),A(d−e))( 𝑥
𝑥′) ≤ 1 with A(e) ∈
ℝm×e, A(d−e) ∈ ℝm×(d−e), x ∈ ℝe, and x′ ∈ ℝd−e. If for each facet F of P at least one
vertex v∉ F survives the projection then the rows of A (d−e) are the Gale transform
of a polytope.
Proof: The rows of the matrix A (d−e) are the Gale transform of a polytope if
for every row ai
(d−e) (i ∈ [m]) the remaining rows of A(d−e) ai
(d−e) are
positively spanning. But for every facet F there exists a vertex v ∉ F that
survives the projection. Hence, the truncated normals corresponding to the
facets containing this vertex positively span ℝd−e. Thus A (d−e) is the Gale
transform of a polytope. So if we project a d-polytope to ℝe such that some
of the vertices survive the projection as described in the above proposition
we obtain a polytope by Gale duality.
Definition 3.1.3 (Associated polytope): Let π be a projection of a d-dimensional
polytope P on m facets to ℝe that preserves one vertex v ∉ F for every facet F of P.
Then the (𝑚 − (𝑑 − 𝑒) − 1) −dimensional polytope on m vertices obtained via
Gale transformation as described in Proposition 3.1.2 is the associated polytope
𝐴(P,π).
Further every face G that is preserved by the projection yields an associated face
AG = [m]HG of the associated polytope 𝐴(𝑃, 𝜋) since Gale duality transforms
positively spanning vectors into faces of the polytope. All these associated faces AG
are simplices. This yields the following subcomplex of the associated polytope.
Definition 3.1.4 (Associated subcomplex): Let π be a projection of a d-
dimensional polytope P on m facets to ℝe that preserves one vertex v∉F for every

37. 37
facet F of P, and let S be the subcomplex of P that is preserved under projection.
Then the associated subcomplex 𝐾(𝑃, 𝜋) is the simplicial complex:
𝐾(𝑃, 𝜋) ∶= {[𝑚] 𝐻 𝐺 | 𝐺 ∈ 𝑆}.
The subcomplex consists of all the facets and their faces.
Now we obtain the following theorem which links the projection of a polytope
preserving certain faces with the embedding of the associated subcomplex into the
associated polytope.
Example 3.1.5:(Projectionof the product of triangles preserving all vertices).
We will use the technique developed in this section to show that there exists no
realization of the product (∆2)2 ⊂ℝ4 of two triangles ∆2 such that the projection
π :ℝ4→ ℝ2 to the plane preserves all 9 vertices.
The product of two triangles is a 4-polytope on 6 facets. Since the projection is to
ℝ2, the associated polytope A((∆2)2,π) is a 3-dimensional polytope. Let us label the
facets of the two triangles by a0,a1,a2 and a′0,a′1,a′2. These are also the labels of the
vertices of the associated polytope A((∆2)2,π). Each vertex of the product lies on
two facets corresponding to two edges of each of the factors. Thus the associated
complex K((∆2)2,π) has an edge for every pair (ai,a′j) with i,j ∈ [3]. So if there
exists a projection of the product of two triangles to the plane preserving all its
vertices, then this yields an embedding of the complete bipartite graph on 3 + 3
vertices K3,3 into the boundary of a 3-polytope. But since K3,3 is not planar there
exists no 3-polytope with K3,3 in its boundary. This implies that there exists no
realization of (∆2)2 such that all vertices survive the projection to the plane. In the
above example we used the non-planarity of the graph K3,3 as a topological

38. 38
obstruction to show that the projection of a product of two triangles to the plane
cannot preserve all the vertices.
3.2 STEREOGRAPHIC PROJECTION
Definition 3.2.1:[9] Let S2 denote the unit sphere x2+y2+z2 =1 in R3 and let N = (0,
0, 1) denote the "north pole" of S2. Given a point M ∈ S2, other than N, then the
line connecting N and M intersects the xy-plane at a point P. Then stereographic
projection is the map
𝜋: 𝑆2
− {𝑁} → 𝐶: 𝑀 ⟼ 𝑃.
Definition 3.2.2: Consider the unit sphere x2 + y2 + z2 = 1 in three dimensions,
capped by the tangent plane z = 1 through the North Pole. We want to defined a
projection from the sphere onto this plane. If P is a point on the sphere, let P' be
the intersection of the ray from the south pole Π = (0, 0, −1) to P with the plane.
This definition fails if P is Π itself. Therefore stereographic projection maps all
points on the sphere except Π to a point on the polar plane, and its inverse wraps
the plane around the complement of Π.
Explicitly, if 𝑃 = (𝑥, 𝑦, 𝑧) 𝑤𝑖𝑡ℎ 𝑧 ≠ −1 then the parameterized line through Π
and 𝑃 𝑖𝑠 𝑃 + 𝑡(𝛱 − 𝑃) = (𝑡𝑥, 𝑡𝑦, 1 − 𝑡 − 𝑡𝑧). This intersects 𝑧 = 1 when
𝑡 =
2
(1 + 𝑧)
which makes

39. 39
𝑃 ′ = (𝑋, 𝑌, 0), 𝑋 =
2𝑥
(1 + 𝑧)
, 𝑌 =
2𝑦
(1 + 𝑧)
Example 3.2.3: Take A = C. Then ℙ1
(ℂ)′
is the set of one-dimensional linear
subspaces of ℂ2. We can choose a unique basis of x ∈ℙ1
(ℂ)′
of the form (1, z)
unless x = (0, z),z ∈ ℂ{0}, and ℂx = ℂ(0,1). In this way we obtain a bijective map
from ℙ1
(ℂ)′
to c∪ {∞}, the extended complex plane. Using the stereographic
projection, we can identify the latter set with a 2-dimensional sphere. The complex
coordinates make it into a compact complex manifold of dimension 1, the Riemann
sphere ℂℙ1
.
Theorem 3.2.4: The image of a straight line in ℂ under stereographic projection is
a circle through 𝑁, with 𝑁 excluded. The image of a circle in ℂ under
stereographic projection is a circle not containing 𝑁. The inverse image of any
circle on 𝑆2 is a straight line together with ∞ if the circle passes through 𝑁,
otherwise a circle.
Proof: Since a straight line in the 𝑥1 𝑥2-plane together with 𝑁 determines a
unique plane, the intersection of which with 𝑆2 is the image of the straight
line we only need to consider the case of a circle in ℂ. If it has center 𝑎 and
radius r its equation is |𝑧 − 𝑎|2
= 𝑟2
or| 𝑧|2
− 2𝑅𝑒( 𝑎̿𝑧) + | 𝑎|2
= 𝑟2
.
Substituting 𝑧 =
𝑥1+𝑖𝑥2
1−𝑥3
into this, using that 𝑥1
2
+ 𝑥2
2
+ 𝑥3
2
= 1 and 𝑥 ≠
1, we get 1 + 𝑥3 − 2𝑥1 𝑅𝑒 𝑎 − 2𝑥2 𝐼𝑚 𝑎 + (1 − 𝑥3)(| 𝑎|2
− 𝑟2) = 0
which is the equation of a plane. [5]Conversely, a circle on the Riemann
sphere is determined by three distinct points. The inverse images of these
three points determine a circle in ℂ. The image of this circle is clearly the
original circle.

40. 40
3.3 ORTHOGONAL PROJECTION
Rendering a picture of a 3-dimensional object on a flat computer screen requires
projecting points in 3-Space to a point in 2-Space. We will discuss only one of
many methods to project points in ℝ3
to points in ℝ2
that preserve the natural
appearance of an object.
Parallel projection simulates the shadow that is cast onto a flat surface by a far
away light source, such as sun. The Figure below shows rays intersecting an object
in 3-Space projection into 2-Space.
The orientation of the axes in the figure above is such that the 𝑥𝑦- plane represents
the computer screen.
To show how to find the 𝑥𝑦 coordinates of the projected point, let the vector
𝑉𝑑 = [
𝑥 𝑑
𝑦 𝑑
𝑧 𝑑
]
represent the direction of the rays. If 𝑥0, 𝑦0, 𝑧0 is a point ℝ3
, then the parametric
equations of the line going through the point and in the direction of 𝑉𝑑 are given by
{
𝑥( 𝑡) = 𝑥0 + 𝑡𝑥 𝑑
𝑦( 𝑡) = 𝑦0 + 𝑡𝑦 𝑑
𝑧( 𝑡) = 𝑧0 + 𝑡𝑧 𝑑
∀ 𝑡 ∈ ℝ
The coordinates of the projection of (𝑥0, 𝑦0, 𝑧0) onto the 𝑥𝑦 plane are found by
letting 𝑧( 𝑡) = 0. Solving for all 𝑡2 we obtain
𝑡 =
−𝑧0
𝑧 𝑑

41. 41
Now, substituting this value of 𝑡 into the first two equations above, we find the
coordinates of the projected point, which are given by
𝑥 𝑝 = 𝑥0 −
𝑧0
𝑧 𝑑
𝑥 𝑑, 𝑦𝑝 = 𝑦0 −
𝑧0
𝑧 𝑑
𝑦 𝑑 𝑎𝑛𝑑 𝑧 𝑝 = 0
The components of 𝑉𝑑 can also be used to find the angles that the rays make with
the 𝑧-axis and 𝑧-plane. In particular, we have
tan φ =
𝑦 𝑑
𝑥 𝑑
𝑎𝑛𝑑 tan 𝜙 =
√ 𝑥 𝑑
2 + 𝑦 𝑑
2
𝑧 𝑑
where 𝜑 is the angle 𝑉𝑑 makes with the 𝑥𝑧 plane and 𝜙 is the angle made with the
𝑧-axis. On the other hand, if the angles 𝜑 and 𝜙 are given, then these equation can
be used to find the projection vector 𝑉𝑑.
3.4 MATRIX PROJECTION
3.4.1 Projection to a Line
[4] Projection matrix 𝑃 projects vector b to a.
Let 𝑃 b= 𝜉a= 𝒑, error 𝑒 = 𝒃 − 𝒑
⟹ 𝐚 ⊥ 𝑒 ⇒ 𝐚T 𝑒 = 0 = 𝐚 𝐓( 𝐛 − 𝒑) = 𝐚 𝐓( 𝐛 − 𝑃𝐛)
⟹ 𝐚 𝐓( 𝐛 − 𝜉𝐚) where 𝜉 is a scalar
⟹ 𝜉𝐚 𝐓
𝐚 = 𝐚 𝐓
𝐛
⟹ 𝜉 =
𝐚 𝐓
𝐛
𝐚 𝐓 𝐚
=
𝐚. 𝐛
𝐚. 𝐚
𝑃𝐛 = 𝜉𝐚 = 𝐚𝜉 = 𝐚
𝐚 𝐓
𝐛
𝐚 𝐓 𝐚
⟹
𝐚𝐚 𝐓
𝐚 𝐓 𝐚
𝐛 ⟹ 𝑃 =
𝐚𝐚 𝐓
𝐚 𝐓 𝐚

42. 42
3.4.2 Projectionto a plane or to an N-dimensional space
Projection matrix 𝑃 projects 𝐛 into the column space A.
Let 𝑃𝐛 = 𝒑 ∈ 𝑪( 𝐀) and 𝐛 − 𝒑 = 𝑒 ∈ 𝑵( 𝐀 𝐓)
𝐀 = [
↑ ↑
𝑎1 𝑎2
↓ ↓
], 𝑃𝐛 = 𝑥1 [
↑
𝑎1
↓
]+ 𝑥2 [
↑
𝑎2
↓
] = 𝐀𝑥̂ = 𝒑, 𝑒 = 𝐛 − 𝒑, 𝐀T
𝑒 = 0
= 𝐀T( 𝐛 − 𝑃𝐛) = 𝐀T( 𝐛 − 𝐀T
𝑥̂) ⟹ 𝐀T
𝐀𝑥̂ = 𝐀T
𝐛
⟹ 𝑥̂ = (𝐀
T
𝐀)
−1
𝐀
T
𝐛 ⟹ 𝑃𝐛 = 𝐀𝑥̂ = 𝐀(𝐀
T
𝐀)
−1
𝐀
T
𝐛
⟹ 𝑃 = 𝐀(𝐀 𝑇
𝐀)−1
𝐀 𝑇
3.4.3 ProjectionMatrix: 𝑷 = 𝑷 𝑻
= 𝑷 𝟐
𝑃𝑃 𝑇
= 𝑃
⟹ 𝑃 =
𝐚𝐚 𝑇
𝐚 𝑇 𝐚
⟹ 𝑃 𝑇
=
( 𝐚𝐚 𝑇) 𝑇
𝐚 𝑇 𝐚
=
( 𝐚 𝑇) 𝑇( 𝐚) 𝑇
𝐚 𝑇 𝐚
=
𝐚𝐚 𝑇
𝐚 𝑇 𝐚
= 𝑃
𝑃 = 𝐀(𝐀 𝑇
𝐀)−1
𝐀 𝑇
⟹ 𝑃 𝑇
= ( 𝐀(𝐀 𝑇
𝐀)−1
𝐀 𝑇) 𝑇
= (𝐀 𝑇
) 𝑇[( 𝐀 𝑇
𝐀)−1] 𝑇
𝐀 𝑇
= 𝐀(𝐀 𝑇
(𝐀 𝑇
) 𝑇
)−1
𝐀 𝑇
= 𝐀(𝐀 𝑇
𝐀)−1
𝐀 𝑇
𝑃2
= 𝑃
⟹ 𝑃 =
𝐚𝐚 𝑇
𝐚 𝑇 𝐚
⟹ 𝑃2
=
𝐚𝐚 𝑇
𝐚 𝑇 𝐚
𝐚𝐚 𝑇
𝐚 𝑇 𝐚
⟹
𝐚( 𝐚 𝑇
𝐚) 𝐚 𝑇
(𝐚 𝑇 𝐚)2
=
𝐚𝐚 𝑇
𝐚 𝑇 𝐚
= 𝑃
⟹ 𝑃 = 𝐀(𝐀 𝑇
𝐀)−1
𝐀 𝑇
⟹ 𝑃2
= ( 𝐀(𝐀 𝑇
𝐀)−1
𝐀 𝑇)( 𝐀(𝐀 𝑇
𝐀)−1
𝐀 𝑇)

43. 43
= 𝐀(𝐀 𝑇
𝐀)−1
𝐼𝐀 𝑇
(since (𝐀 𝑇
𝐀)(𝐀 𝑇
𝐀)−1
= 𝐼)
𝐀(𝐀 𝑇
𝐀)−1
𝐀 𝑇
= 𝑃
3.4.5 Example 1:
Find the projection of vector [
1
2
] on vector [
1
−1
]
𝑃 =
[1 −1][
1
2
]
[1 −1][
1
−1
]
[
1
−1
] =
−1
2
[
1
−1
] = [
−1
2⁄
−1
2⁄
]
Find the projection matrix 𝑃 that projects any given vector in ℝ2
to the
vector [
1
−1
].
𝑃 =
[
1
−1
][1 −1]
[1 −1][
1
−1
]
= 1
2⁄ [
1 −1
−1 1
]
3.4.6 Example 2:
Find (i) the projection of vector 𝐛 = [
1
1
1
] on the column spaceof matrix 𝐀 =
[
1 0
1 1
0 1
] (ii) the projection matrix 𝑃 that projects any vector ℝ3
to 𝐶(𝐀)
⟹ ( 𝑎) Determine the coefficient vector 𝑥̂ based on 𝐀 𝑇
e = 0, then determine 𝒑
from 𝒑 = 𝐀𝑥̂.
𝐀 𝑇
e = 0 = 𝐀 𝑇( 𝐛 − 𝒑) = 𝐀 𝑇( 𝐛 − 𝐀𝑥̂)

44. 44
⟹ 𝐀 𝑇
𝐛 = 𝐀 𝑇
𝐀𝑥̂ ⟹ 𝑥̂ = ( 𝐀 𝑇
𝐀)−1
𝐀 𝑇
𝐛
= ([
1 1 0
0 1 1
][
1 0
1 1
0 1
])
−1
[
1 1 0
0 1 1
][
1
1
1
] = ([
2 1
1 2
])
−1
[
2
2
] =
1
3
[
2 −1
−1 2
][
2
2
]
=
1
3
[
2
2
]
⟹ 𝒑 = 𝐀𝑥̂ = [
1 0
1 1
0 1
]
1
3
[
2
2
] =
1
3
[
2
4
2
] =
[
2
3⁄
4
3⁄
2
3⁄ ]
⟹ ( 𝑏) Find the projection matrix 𝑃 first, then determine 𝒑 from 𝒑 = 𝑃𝐛
𝑃 = 𝐀(𝐀 𝑇
𝐀)−1
𝐀 𝑇
= [
1 0
1 1
0 1
]([
1 1 0
0 1 1
][
1 0
1 1
0 1
])
−1
[
1 1 0
0 1 1
]
= [
1 0
1 1
0 1
]((
1
3
)[
2 −1
−1 2
])[
1 1 0
0 1 1
]
1
3
[
2 −1
1 1
−1 2
][
1 1 0
0 1 1
] =
1
3
[
2 1 −1
1 2 1
−1 1 2
]
3.5 APPLICATION OF MAP PROJECTION TO THE WORLD

45. 45
3.5.1 Orthographic
The orthographic projection is how the earth would appear if viewed from a distant
planet. Since the light source is at an infinite distance from the generating globe,
all rays are parallel. This projection appears to have been first used by astronomers
in ancient Egypt, but it came into widespread use during World War II with the
advent of the global perspective provided by the air age. It is even more popular in
today's space age, often used to show land-cover and topography data obtained
from remote sensing devices. The generating globe and half-globe illustrations in
this bookare orthographic projections, as is the map on the front cover of the book.
The main drawback of the orthographic projection is that only a single hemisphere
can be projected. Showing the entire earth requires two hemispherical maps.
Northern and southern hemisphere maps are commonly made, but you may also
see western and eastern hemisphere maps.
3.5.2 Stereographic
Projecting a light source from the antipodal point on the generating globe to the
point of tangency creates the stereographic projection. This is a conformal
projection, so shape is preserved in small areas. The Greek scholar Hipparchus is
credited with inventing this projection in the second century BC. It is now most
commonly used in its polar aspect and secant case for maps of polar areas. It is the
projection surface used for the Universal Polar Stereographic grid system for polar
areas, as we will see in the next chapter. A disadvantage of the stereographic:
conformal projection is that it is generally restricted to one hemisphere. If it is not
restricted to one hemisphere, then the distortion near the edges increases to such a
degree that the geographic features in these areas are basically unrecognizable. In
past centuries, it was used for atlas maps of the western or eastern hemisphere.

46. 46
3.5.3 Azimuthal equidistant
The azimuthal equidistant projection in its polar aspect has the distinctive
appearance of a dart board—equally spaced parallels and straight-line meridians
radiating outward from the pole. This arrangement of parallels and meridians
results in all straight lines drawn from the point of tangency being great circle
routes. Equally spaced parallels mean that great circle distances are correct along
these straight lines. The ancient Egyptians apparently first used this projection for
star charts, but during the air age it also became popular for use by pilots planning
long-distance air routes. In the days before electronic navigation, the flight
planning room in major airports had a wall map of the world that used an oblique
aspect azimuthal equidistant projection centered on the airport. You will also find
them in the public areas of some airports. All straight lines drawn from the airport
are correctly scaled great circle routes. This is one of the few planar projections
that can show the entire surface of the earth.
3.5.4 Lambert Azimuthal Equal Area
In 1772 the mathematician and cartographer Johann Heinrich Lambert published
equations for the tangent case planar Lambert azimuthal equal area projection,
which, along with other projections he devised, carries his name. This planar equal
area projection is usually restricted to a hemisphere, with polar and equatorial
aspects used most often in commercial atlases. More recently, this projection has
been used for statistical maps of continents and countries that are basically circular
in overall extent, such as Australia, North America, and Africa. You will also see
the oceans shown on maps that use the equatorial or oblique aspects of this
projection. The Lambert azimuthal equal area projection is particularly well suited
for maps of the Pacific Ocean, which is almost hemispheric in extent.

47. 47
CHAPTER FOUR
CONCLUSION

48. 48
In this project work, we have been able to see the introduction, concept and
relation to real life of map projections. We have also seen some of the properties of
map projections and the some applications of map projection in relation to
different aspects of mathematics and some of its applications to the real world.
REFERENCES
[1]C. Adams and R. Franzosa, Introduction to Topology, Pearson (2006)

49. 49
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