Principles Of Mathematical Analysis Chapters 1,2,3 Rudin

Front 1

Order

Back 1

Let S be a set. An order on S is a relation denoted by <,with the following two properties:
(i)If x Ɛ S and y Ɛ S then one and only one of the statements is true:
x<y ,x=y, y<x
(ii)If x,y,z Ɛ S if x<y and y<z then x<z

Front 2

Ordered Set

Back 2

An ordered set is a set S in whach and order is defined.

For Example, Q is an ordered set if r<s is defined to mean that s-r is a positive rational number

Front 3

Bound Above/Upper Bound

Back 3

Suppose S is an ordered set, and E ⊂ S. If there exists B e S such that x ≤ B for every x Ɛ E, we say that E is bounded above and call B an upper bound of E

Front 4

Bound Below/Lower Bound

Back 4

Suppose S is an ordered set, and E ⊂ S. If there exists B e S such that x ≥ B for every x Ɛ E, we say that E is bounded below and call B a lower bound of E

Front 5

Greatest Lower Bound/Infimum

Back 5

The Infimum of set E which is bounded below is defined as:

a=inf E means that a is a lower bound of E and that no B with B >a is a lower bound of E

Front 6

Least Upper Bound/Supremum

Back 6

The Supremum of set E which is bounded below is defined as:

a=Sup E means that a is a upper bound of E and that no B with B <a is a upper bound of E

Front 7

Least-upper-bound property

Back 7

An ordered set S is said to have the least-upper-bound property if the following is true:
If E ⊂ S, E is not empty, and E is bounded above, then sup E exists in S.

Front 8

Field

Back 8

A field is a set F with two operations called Addition and Multiplication which satisfy the so called "field Axioms", (A) (M) (D)

Front 9

Field Axioms (A)

Back 9

(A1) If x Ɛ F and y Ɛ F, then there sum x+y is in F.
(A2)Addition is commutative
(A3) Addition is associative
(A4) F contains an element 0 such that 0+x=x for every x Ɛ F
(A5) to every x Ɛ F corresponds to an element -x Ɛ F such that x+(-x)=0

Front 10

Field Axioms (M)

Back 10

(M1) If x Ɛ F and y Ɛ F then their product xy is in F
(M2) Multiplication is commutative
(M3) Multiplication is assiociative
(M4) F contains an element 1 ≠ 0 such that 1x=x for every x Ɛ F.
(M5) If x Ɛ F and x ≠0 then there exists an element 1/x Ɛ F such that x*(1/x)=1

Front 11

Field Axioms(D)

Back 11

The distributive law:

x(y+z)=xy+xz
holds for all x,y,z Ɛ F

Front 12

Ordered Field

Back 12

An ordered field is a field F which is also and ordered set such that
(i) x+y<x+z if x,y,z Ɛ F and y<z
(ii) xy>0 if x Ɛ F,y Ɛ F, and x>0 and y>0

Front 13

Real Field

Back 13

There exists and ordered field R which has the least upper bound property.
Moreover, R contains Q as a subfield

Front 14

Extended Real Number System

Back 14

The extended real number system consists of the real field R and two symbols +∞ and -∞. We preserve the original order in R and define -∞<x<+∞ for every x Ɛ R

Front 15

Complex Number

Back 15

A complex number is an ordered pair (a,b) or real numbers. "Ordered" means that (a,b) and (b,a) are regarded as distinct if a ≠ b.

Let x=(a,b) and y=(c,d) we say x=y if a=c and b=d. We define:
x+y=(a+c,b+d)
xy=(ac-ba,ad+bc)

Front 16

Complex Field

Back 16

The set of all complex numbers will (0,0) as 0 and (1,0) as 1. The (A) and (M) and (D) axioms hold

Front 17

Euclidean Space

Back 17

The Euclidean Space is defined by having the vector space R^k with the inner product and the norm defined.

Inner product: x dot y= ∑xi*yi

Norm: |x|=(x*x)^.5=(∑xi^2)^.5

Front 18

Map

Back 18

Consider two sets A and B, whose elements may be any objects whatsoever and suppose that with each element x of A there is associated , in some manner an element of B, which we denote by f(x). Then f is said to be a function from A to B( or a mapping of A into B)

Front 19

Domain/Range

Back 19

Consider two sets A and B, whose elements may be any objects whatsoever and suppose that with each element x of A there is associated , in some manner an element of B, which we denote by f(x). The set A is called the domain of f( we also say f is defined on A and the elements f(x) are called the values of f. The set of all values of f is called the range of f.

Front 20

Surjective

Back 20

In mathematics, a function f is said to be surjective or onto, if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f(x) = y .

Said another way, a function f: X → Y is surjective if and only if its range f(X) is equal to its codomain Y. A surjective function is called a surjection.

Front 21

Injective

Back 21

In mathematics, an injective function is a function which associates distinct arguments with distinct values; that is, every unique argument produces a unique result.

An injective function is called an injection, and is also said to be a one-to-one function (not to be confused with one-to-one correspondence, i.e. a bijective function).

A function f that is not injective is sometimes called many-to-one. (However, this terminology is also sometimes used to mean "single-valued", i.e. each argument is mapped to at most one value.)

Front 22

Bijective

Back 22

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that
f(x) = y.

Alternatively, f is bijective if it is a one-to-one correspondence between those sets; i.e., both one-to-one (injective) and onto (surjective). (One-to-one function means one-to-one correspondence (i.e., bijection) to some authors, but injection to others.)

Front 23

Equivalence Relation

Back 23

If there exists a 1-1 mapping of A onto B, we say that A and be can be put in 1-1 correspondence or that A and B have the came cardinal number,or briefly that A and B are equivalent and we write A~B. It has the following properties
(i) It is reflexive A~A
(ii) Symmetric If A~B then B~A
(iii)Transitive if A~B and B~C then A~C

Front 24

Finite

Back 24

For any positive integer n, let Jn be the set whose elements are the integers 1,2,..,n;let J be the set consisting of all positive integers for any set A we say:
A is finite if A~Jn for some n (the empty set is said to be finite)

Front 25

Infinite

Back 25

For any positive integer n, let Jn be the set whose elements are the integers 1,2,..,n;let J be the set consisting of all positive integers for any set A we say:
A is infinite if A is not finite

Front 26

countable

Back 26

For any positive integer n, let Jn be the set whose elements are the integers 1,2,..,n;let J be the set consisting of all positive integers for any set A we say:
A is countable if A~J

Front 27

Uncountable

Back 27

For any positive integer n, let Jn be the set whose elements are the integers 1,2,..,n;let J be the set consisting of all positive integers for any set A we say:
A is uncountable if A is neither finite nor countable

Front 28

At Most Countable

Back 28

For any positive integer n, let Jn be the set whose elements are the integers 1,2,..,n;let J be the set consisting of all positive integers for any set A we say:
A is at most countable if A is finite or countable

Front 29

Sequence

Back 29

By a sequence we mean a function f defined on the set J of all positive integers. If f(n) =xn for n Ɛ J it is customay to denote the sequence f by the symbol {xn} or sometimes x1,x2,x3...The values of f, that is, the elements xn, are called terms of the sequence. If A a set and if xn Ɛ A for all n Ɛ J then {xn} is said to be a sequence in A or a sequence of elements of A.

Front 30

Union

Back 30

Let A and K be sets, and suppose that with each element a of A there is associated a subset of K which we denote by Ea

The union of sets Ea is defined to be the set S such that x Ɛ S if and only if x Ɛ Ea for at least one a Ɛ A

Front 31

Intersection

Back 31

Let A and K be sets, and suppose that with each element a of A there is associated a subset of K which we denote by Ea

The intersection of the sets Ea is defined to be the set P such that x Ɛ P if and only if x Ɛ Ea for every a Ɛ A.

Front 32

Metric Space/Metrics

Back 32

A set X,whose elements we shall call points is said to be a metric space if with any two point p and q of X there is associated a real number d(p,q), called the distance from p to q such that
(i) d(p,q)>0 if p≠0;d(p,p)=0
(ii) d(p,q)=d(q,p)
(iii) d(p,q) ≤ d(p,r)+d(r,q)
any function with these three properties is called a distance function or metric

Front 33

k-cell

Back 33

If ai<bi for i=1,...,k the set of all points x=(x1,...,xk) in Rk whose coordinates satisfy the inequalities ai≤xi≤bi(1≤i≤k) is called a k-cell
Thus a 1-cell is an interval and a 2-cell is a rectangle

Front 34

Open/Closed Ball

Back 34

If x Ɛ Rk and r>0 the open(closed) ball B with a center at x and radius r is defined to be the set of all y Ɛ Rk such that |y-x|<r for an open ball and |y-x|≤r for a closed ball

Front 35

Convex

Back 35

We call a set E ⊂ Rk convex if
λx+(1-λ)y Ɛ E
whenever x Ɛ E, y Ɛ E and 0<λ<1

Front 36

Neighborhood

Back 36

A neighborhood of a point p is a set Nr(p) consisting of all points q such that d(p,q)<r. The number r is the radius of Nr(p).

Front 37

Limit Point

Back 37

A point p is a limit point of the set E if every neighborhood of p contain a point q≠p such that q Ɛ E

Front 38

Isolated Point

Back 38

If p Ɛ E and p is not a limit point of E, then p is called an isolated point of E

Front 39

Interior Point

Back 39

A point p is an interior point of E if there is a neighborhood N of p such that N ⊂ E

Front 40

Closed/Open

Back 40

E is closed if all its limit points are contained within E

E is open if every point of E is an interior point of E

Front 41

Complement

Back 41

The complement of E is the set of all points p Ɛ X such that p ∉E

Front 42

Perfect

Back 42

E is perfect if E is closed and if every point of E is a limit point of E

Front 43

Bounded

Back 43

E is bounded if there is a real number M and a point q Ɛ X such that d(p,q)<M for all p Ɛ E

Front 44

Dense

Back 44

E is dense in X if every point of X is a limit point of E, or a point of E(or both)

Front 45

Closure of a Set

Back 45

If X is a metric space, if E ⊂ X and if E' denoted the set of all limit point of E in X, then the closure is the set Ē=E U E';

Front 46

Open Relative to a Subset

Back 46

Suppose E⊂Y⊂X where X is a metric space. To say that E is an open subset of X mean that to each point p Ɛ E there is associated a positive number r such that the condition d(p,q)<r, q Ɛ X imply that q Ɛ E. But we have already observed that Y is also a metric space ,so that our definitions may equally well be made within Y. To be quite explicit,let us say that E is open relative to Y if to each p Ɛ E there is associated r>0 such that q Ɛ E whenever d(p,q)<r and q Ɛ Y.

Front 47

Open Cover

Back 47

By an open cover of a set E in a metric space X we mean a collection {Ga} of open subsets of X such that E ⊂ Ua Ga

Front 48

Compact Set

Back 48

A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover

Front 49

Cantor Set

Back 49

the set obtained from the closed interval from 0 to 1 by removing the middle third from the interval, then the middle third from each of the two remaining sets, and continuing the process indefinitely.Which constructs a perfect and uncountable set.

Front 50

Separated/Connected

Back 50

Two subsets A and B of metric space X are said to be separated if both A ∩ B(closure) and A(closure) ∩ B are empty. A set E ⊂ X is said to be connected if E is not a union of two nonempty separated sets.

Front 51

Convergent Sequence

Back 51

A sequence {pn} in a metric space X is said to converge if there is a point p ∈ X with the following property: For every ε>0 there is an integer N such that n≥N implies that d(pn,p)< ε.
In this case we also say that {pn} converges to p, or that p is the limit of {pn} and we write pn-->p or lim pn=p;

Front 52

Divergent Sequence

Back 52

If {pn} does not converge it is said to diverge.

Front 53

Subsequence

Back 53

Given a sequence {pn} consider a sequence {nk} of positive integers such that n1<n2<n3...Then the sequence {pni} is called a subsequence of {pn}.

Front 54

Subsequencial Limit

Back 54

Given a sequence {pn} consider a sequence {nk} of positive integers such that n1<n2<n3...Then the sequence {pni} is called a subsequence of {pn}.

If {pni} converges, its limit is called a subsequential limit of {pn}

Front 55

Cauchy Sequence

Back 55

A sequence {pn} in a metric space X is said to be a Cauchy sequence if for every ε>0 there is an integer Nsuch that d(pn,pm) < ε if n≥N and m≥N

Front 56

Diameter

Back 56

Let E be a subset of metric space X and let S be the set of all real numbers of the form d(p,q), with p ∈ E and q ∈ E. The sup of S is called the diameter of E

Front 57

Complete Metric Space

Back 57

A metric space in which every cauchy sequence converges

Front 58

Monotonically Increasing

Back 58

A sequence {sn} of real numbers is said to be monotonically increasing if
sn≤sn+1 (n=1,2,3,...)

Front 59

Monotonically decreasing

Back 59

A sequence {sn} of real numbers is said to be monotonically decreasing if
sn≥sn+1 (n=1,2,3,...)