**Chapter 9
**** Abstract algebra ****h**** linear space ****h**** functional analysis**

This chapter includes three parts: abstract algebra, linear space and functional analysis, focusing on linear space . In order to introduce the needs of linear space, here is a brief introduction to the basic knowledge of abstract algebra, that is, basic concepts such as groups, rings, fields, etc. Simple properties . Functional analysis is introduced as an example of the analytical application of the theory of linear spaces, so it will not be described systematically . In addition to describing the basic concepts and important properties of Lebesgue integrals, a brief introduction The normed linear space, Hilbert space, Banach space and some simple properties of them are introduced .

In the part of linear space, the definitions, properties and some algorithms of linear space, linear transformation, unitary space, quadratic form and Hermitian form, and standard form of square matrix are introduced .

**§ ****1 ****Abstract Algebra**

1. Basic Algebra System

[ Algebraic operations ] Assume that for a set (see Chapter 21, § 1 , 1) , any element a in *A* and any element *b* in set *B* can correspond to *a* uniquely determined element c in *a set **C* according to a certain rule , then this corresponds to a (binary) algebraic operation of A and B. The sets *A* and *B* can *also* be *the* same set, that is, for any two elements *a* , *b in **A , the element **c* can be uniquely determined , so that , *c* Can belong to *A* or not belong to *A* , if it belongs to *A* , then *A* is said to be closed under the operation ._{}_{}

Under binary operations , if it holds for any two elements *a* and *b of **A* , then *A* is commutative . If it holds for any three elements *a* , *b* , and *c of **A* , then *A* is said to be commutative. Associative . If the operation is an ordinary addition or multiplication, it is written as or respectively . Both addition and multiplication in the set of integers are commutative and associative, so the set of integers is commutative and associative ._{}_{}_{}_{}_{}_{}_{}

[ Algebraic system ] If a set *A **has* algebraic operations that satisfy certain laws, it is called an algebraic system . Groups, rings, and fields are three basic algebraic systems .

Second, the group

[ Definition and Examples of Groups ] Suppose *G* is not an empty set (see Chapter 21, § 1 , 1), and an algebraic operation is given to *G.* If the following four conditions are satisfied, then *G* is called a group:_{}_{}

( i ) *G* is closed below, that is, for each pair of elements , there is a unique determinate element , and . _{}_{}_{}_{}

( ii ) *G* is associative below, that is, for any , we have _{}_{}

_{}

( iii ) There is an element *e* in *G* , for either , satisfying _{}

_{}

( iv ) For either , there is one , satisfying _{}_{}

_{}

*e* in condition ( iii ) is called the identity element or identity element; in condition ( iv ) is called the inverse element ._{}_{}

Note that the condition ( iii ) in the definition can be changed to: there is a left identity element *e* (or right identity element ), such that (or ), holds for any pair . Because it is deduced from this . Therefore, the identity element in the group is unique .
Definition The middle condition ( iv ) can be changed to: each element has a left (or right) inverse , which makes (or ) true . Because it follows from this , it also holds . Therefore, the inverse of each element in the group is unique ._{}_{}_{}_{}_{}_{}_{}_{}_{}_{}_{}

If the multiplication of a group *G* is commutative, then *G* is called a commutative group or an Abelian group .
Especially under addition, the commutative group is called an additive group . In addition , the inverse element is changed to a negative element - , The unit element is called the zero element, denoted as 0.
_{}_{}_{}_{}_{}

Example 1
The set of integers *N* forms an additive group; the set of rational numbers, the set of real numbers, and the set of complex numbers each form an additive group .

Example 2
The set of nonzero real numbers forms a group for multiplication . The set of positive real numbers also forms a group for multiplication ._{}_{}

Example 3
The *nth* -order invertible matrices with all elements in the number field *F* form a group for the multiplication of matrices, denoted as ._{}

Example 4
Suppose Ω is a plane figure, which is the set of all orthogonal transformations that make Ω immobile on the plane, then it forms a group . It is generally called the symmetry group of graph Ω ._{}_{}_{}

Example 5
The set of all *n* permutations forms a group, called the permutation group, denoted by ._{}

In fact, if any two permutations are taken *n times:*

_{}

_{}It can be rewritten as: for the permutation sum , the permutation and their correspondence are specified, that is , the product of the sum, denoted as under this multiplication, it is not difficult to deduce that the conditions specified in the group are satisfied, thus forming a group .

_{}

_{}_{}

_{}

_{}_{}_{}

_{}

_{}_{}

Example 6 All the reversible transformations of a non-empty set *S* to itself (see Chapter 21, § 1 , 2) for the multiplication of transformations form a group, called the group of total transformations of the set *S* , denoted as a subgroup of . The transformation group on *S.*_{}_{}

[ Basic properties of groups ]

1 ^{o} In a group, for any element *a* , *b* , each equation has a solution . That is . 2 ^{o} The elimination law holds . That is, if , then . 3 ^{o} The general associative law holds in the group . That is, 4 ^{o} The general commutative law holds in the commutative group .That is , any permutation of is in the formula .

_{}

_{}

^{} _{}_{}

^{}

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^{}

_{}

_{}_{}

[ Subgroup ] If a non-empty subset *H* of a group *G* also forms a group for the operations of G, then *H **is* called a subgroup of *G.*

A sufficient and necessary condition for a non-empty subset *H* of a group *G* to be a subgroup is: If , then ._{}_{}

The intersection of any number of subgroups (see Chapter 21, § 1 , 3) is a subgroup .

[ Cyclic group ] The totality of all powers of an element *a* forms a group, which is called a cyclic group .
A cyclic group is a commutative group ._{}

If no two elements in the sequence are equal, *G* is called an infinite cyclic group . If there are equal elements, then_{}

_{}

It can be deduced that *G* is a set of *n* elements , that is, *G* is called a finite cyclic group at this time, and *n is* called the order of *G , that is, **n* is the smallest positive integer that makes ._{}

_{}

_{}

A subgroup of a cyclic group is also a cyclic group .

[ Invariant subgroup · coset · quotient group ] Let *H* be a subgroup of group *G* , if for each element , there is
( here means the product of *g* and all elements in *H* , for example ), that is , *H* is called An invariant subgroup (or normal subgroup) of
*G.* and are called the left and right cosets of *G* for *H* with element *g* , respectively . Hence the left and right cosets of an invariant subgroup with the same element are coincident ._{}

_{}_{}_{}_{}_{}_{}

When cosets are regarded as elements, all cosets form a group, which is called the quotient group of *G* to *H* , denoted as *G* / *H* .

Lagrange's theorem The order of a subgroup of a finite group *G* is a factor of the order of the group *G.*

*The order of the quotient group of the invariant subgroup **H of **G* is the quotient obtained by dividing the order by the order of ._{}_{}_{}

All subgroups of a commutative group are invariant subgroups .

*A* group *G* is said to be simple if it has no invariant subgroups other than itself .

[ Isomorphism and Automorphism ] Set up two groups , if the product of any two elements *a* and *b* in , corresponds to the product of the corresponding elements in , and only corresponds to this product, that is,_{}_{}_{}

_{}

The one- to-one correspondence to the above with this property is called an isomorphism, also known as an isomorphism with and , denoted as . The isomorphism of a group *G* to itself is called an automorphism ._{}_{}_{}_{}_{}

Isomorphism has the following properties:

1 ^{oUnder} isomorphism, the identity element, inverse element, and subgroup of one group correspond to the identity element, inverse element, and subgroup of another group, respectively .

2 ^{o} isomorphism is an equivalence relation, that is

(i) reflexivity ; _{}

(ii) symmetry if , then ; _{}_{}

(iii) Transitivity If , , then . _{}_{}_{}

3 ^{o} Cayley's Theorem Any group *G* is isomorphic to a certain transformation group of its element set . * *

[ Homomorphism and Automorphism ] There are two groups , , and a mapping : . Let , if_{}_{}_{}_{}_{}

_{}

Then it is called a homomorphism . A homomorphism of , denoted as ~ . The homomorphism of a group to itself is called an autohomomorphism ._{}_{}_{}_{}_{}_{}

Homomorphism has the following properties:

1 ^{o} A one-to-one homomorphism is an isomorphism .

2 ^{o} Under homomorphism, the identity element maps to the identity element, and the inverse element maps to the inverse element .

3 ^{o} Assuming that *f* is a homomorphism of the group G *, then the set **N* formed by all elements of the corresponding identity element in *G* is an invariant subgroup of *G. **N* is called the kernel of the homomorphism *f* , denoted as ._{}_{}_{}_{}

4 ^{o} Assuming that the group *G* is homomorphic, then the set of all elements corresponding to any fixed element in G is a coset of *G* to the *homomorphic* kernel *N.*_{}_{}

5 ^{o} The fundamental theorem of homomorphism assumes that *G* , a homomorphism, a one-to-one correspondence between the elements of the coset of the group *G* to *N* and the quotient group *G* / *N* is an isomorphism . It shows that the homomorphism of *G* is the same as The corresponding quotient group *G* / *N* is isomorphic . _{}_{}_{}_{}

Third, the ring

[ Definition and Examples of Rings ] A non-empty set *R* has two binary operations, addition and multiplication. If the following three conditions are satisfied, then *R* is called a ring:

( i ) *R* is an additive group;

( ii ) satisfy the associative law for multiplication . That is, for any , there is_{}

_{}

( iii ) The addition and multiplication satisfy the left and right distributive laws . That is , for any ring, if it satisfies the commutative law of multiplication , then *R* is called a commutative ring ._{}

_{}

_{}

Example 1 The whole of all integers is a ring, called the ring of integers .

Example 2
Suppose *F* is a number field, then the totality of polynomials on the field *F is a ring, denoted as **F* [ *x* ].

Example 3
If the sum, difference and product of any two numbers in the number set *R* still belong to *R* , then *R* is also a ring, called a number ring . A single number zero is also a number ring, called a zero ring. Obviously, the number ring is always exchange ring .

Example 4
If *R* is a ring, all the *n -order square matrices formed by the elements of **R* form a ring under the addition and multiplication of the matrix, which is called the *n* -order full square matrix ring on *R* , denoted as . At that time , which is a non-commutative ring ._{}_{}_{}

[ Basic properties of rings ] Because a ring is an additive group, it has all the properties of an additive group . Therefore, only the properties represented by multiplication are introduced .

1 ^{o}_{}

^{2o} _^{}_{}

3 ^{o} holds for the distributive law of subtraction, that is

_{}

4 ^{o} The general associativity law holds, that is

_{}

5 ^{o} The general distributive law holds, that is

_{}

6 ^{o} For any integer *m* , we have

_{}

7 ^{o} The exponential law for positive integers holds, that is, for commutative rings there are

_{}

_{}

[ Zero factors and identity elements ] In the ring *R* , if , then *a* is called the left (right) zero factor of *R* , denoted as . A is also called the left (right) zero element *of b* . *An* element is also The left and right zero factors are called zero factors . If there are no zero factors in the ring, it is called a zero factor ring . The *n* -order full square matrix ring is a zero factor ring ._{}_{}_{}

If there is an element in the ring *R* , and for any one , it is called the left (right) identity element of *R.* If it is the left and right identity element at the same time, that is , *e* is called the identity element of *R.* At this time, *R* is called as There is an identity ring . For example, an integer ring is an identity ring, and 1 is its identity element; an *nth* -order full square matrix ring has an identity element , which is the identity matrix *I.*_{}_{}_{}_{}_{}_{}_{}

If *R* has an identity element, the identity element is unique; if *R* has an identity element *e* , and the pair has an inverse element , it is unique ._{}_{}_{}

A commutative ring with a unit element but no zero factor is called an integral ring . For example, a ring of integers and a number field are all integral rings .

[ Subring and Expansion Ring ] Let *S* be a subset of the ring *R , if the two operations of **S* on *R* form a ring, then *S* is called a subring of *R , and **R* is called the expansion ring of *S.*

The ring itself can be regarded as its sub-ring, and the zero-ring is also its sub-ring . The sub-ring which is different from itself and the zero-ring is called the true sub-ring .

A sufficient and necessary condition for a subset *S* of a ring *R to be a subring of **R is:*

(i) *S* is a non-empty set;

(ii) if , then ;_{}_{}

(iii) If , then ._{}_{}

[ Ideal and principal ideal ] Let *R* be a ring and *I* be a subset of R. If *the* difference between any two elements in *I* and the sum of the products of any element *a in **I* and any element *r in **R* belong to *I* , then it is said that *I* is an ideal of *R.* For example, the set of even numbers is an ideal of a ring of integers . Every ideal is a subring of a given ring, the inverse of which is not true ._{}_{}

The intersection of any ideals of a ring is still the ideal of the ring . In particular, the intersection of all ideals containing a fixed element *r* in the ring is still the ideal of the ring, that is, it is an ideal generated by an element *r* , which is called The main ideal, denoted by ( *r* ) .

4. Domain

[ Definition and Examples of Fields ] A commutative ring *R* with identity elements is called a field if it contains at least one non-zero element, and every non-zero element *a* always has an inverse *.*_{}

Example 1
The number field *F* (the rational number field **Q** , the real number field **R** , the complex number field **C** , etc.) are all fields .

Example 2
All rational fractions ( and ) on the number field *F* form a field under the addition and multiplication of rational fractions, which is called the field of rational fractions on the number field *F.*_{} _{}_{}

[ Basic properties of domains ]

1 ^{The o} domain has no zero factor .

2 ^{o }*A* set *F* is a field if it satisfies the following conditions under two binary operations (addition and multiplication) :

(i) *F* is the additive group with zero as the unit element;

(ii) the set consisting of all elements of *F* except zero is a commutative group under multiplication;

(iii) Multiplication is assignable to addition, ie ._{}

3 ^{oIn} the field *F* , the equation ( , and ) has a unique solution and can be written as ._{}_{}_{}_{}

4 ^{o} In the domain *F* , the exponential law holds:

_{}

where *m* , *n* are any integers, *a* , *b* are any two elements in *F* , and only non-zero elements can have negative integer powers .

5 ^{o} If *n* times the unit *e* of the field *F* is abbreviated as *n* , then the *n* times of any element *a in **F* is the product of *n* and *a* ._{}_{}_{}

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