**§ ****7 ****Preliminary functional analysis**

The
Lebesgue integral

1. Measures and Measurable Functions

[ Measurement and measurable set ] Let *S* be any set of bounded points in a certain interval , then the infimum of the sum of the lengths of a set of intervals covering *S* is called the outer measure of S, denoted *as* . Contains *S* The difference between the length of any bounded interval of and the outer measure of the complement of *S* (that is, the entire set of points that do not belong to *S* ) is called the inner measure of S, and the set S denoted by *.* = is *called* the measurable set , whose measure is denoted as ._{}_{}_{}_{}_{}_{}_{}_{}_{}_{}

Let *S* be an unbounded point set on a straight line. If all *x* greater than zero is measurable , then this unbounded point set *S* is called measurable . In this case, the measure of the unbounded point set *S is defined as*_{}

_{}

This can be limited or unlimited ._{}

Every bounded open set is measurable .

The concept of measurable sets can be extended to point sets in *n* -dimensional spaces .

[ Almost everywhere ] A property is said to hold almost everywhere on a given interval if it holds at all points on the interval except for a set of measure zero .

[ Measurable function ] Suppose a function is defined on a measurable set *S* , and *c* is any real number . If the set formed by all points *x* on *S* is measurable, then the function is called a measurable function on *S* function ._{}_{}_{}

In this definition, the condition can be replaced by any one of , , ._{}_{}_{}_{}

Any continuous function within is a measurable function within ._{}_{}

If both are inner measurable functions, then ( *a* is a constant), and (the limit exists) are also inner measurable functions ._{}_{}_{}_{}_{}_{}

2. Lebesgue integral

[ Lebesgue integral of a bounded function ] Given a bounded and measurable real function in a bounded interval , insert bisectors within the variation of ( ):_{}_{}_{}_{}

_{} ( 1 )

And use the set formed by the point *x* to represent that , for each sequence of divisions ( 1 ) , at that time , the sum tends to a unique finite limit *I* , denoted as_{}_{}_{}_{}_{}_{}

_{}

This quantity is called a definite integral in the Lebesgue sense, also known as a Lebesgue integral, and is said to be integrable in the interior ._{}_{}_{}_{}

[ Lebesgue integral of unbounded function ] If it is an unbounded measurable function in a bounded interval , the Lebesgue integral is defined as follows:_{}_{}_{}

_{}

in the formula

_{}

[ Lebesgue integral over unbounded interval ] If exists for everything , then define the Lebesgue integral as follows:_{}_{}_{}

_{}

in the formula _{}

It is also possible to define and ._{}_{}

[ Lebesgue integral over a set of points ] The above definition of the Lebesgue integral of bounded and unbounded functions can be extended to the Lebesgue integral over any measurable set S. It can *also* be extended to the region of *n* -dimensional space or multiple Lebesgue integrals over measurable sets ._{}

[ Existence and Properties of Lebesgue Integrals ]

1 ^{o} Every bounded measurable set function is integrable over any bounded measurable set, and an integrable function over a measurable set *S* is integrable over every subset of *S.*

2 ^{o} The necessary and sufficient conditions for the existence of Lebesgue integrals are: the existence of Lebesgue integrals ._{}_{}

3 ^{o} The Lebesgue integral over a set with measure equal to zero is equal to zero .

4 ^{o} Let o be a set of countable disjoint (i.e. ) measurable sets, assuming that Lebesgue integrals on and on both exist, then_{}_{}_{}_{}_{}

_{}

5 ^{o} Continuity Theorem Suppose that a positive function is measurable on a measurable set *S* , and for all *n* and all *x in **S* , the inequality _{}_{}

_{}

It is established almost everywhere; and let it be established for almost all *x* in *S* , then_{}

_{}

exists, and

_{}

6 ^{o} Lebesgue's Fundamental Theorem Suppose *S* is a measurable set, not necessarily bounded . _{}

like

(i) _{}are all non-negative measurable functions on *S ;*

(ii)_{}

but
_{}

3. Square integrable function

[ *L *_{2} space ] If *S* is a bounded measurable set, *f* ( *x* ) is a measurable function on *S* , integrable, and_{}

_{}

is called a function belonging to space, denoted as , or abbreviated as . In this paragraph, it is assumed that *S* is the interval ._{}_{}_{}_{}_{}

If , , then both are integrable; and there is_{}_{}_{}

_{}

[ modulus and distance ] is set , then it is called_{}

_{}

is the norm (norm) of *f* .

designate __{}

_{}

is the distance between *f* and *g* .

set rules_{}

(i) _{}, only if it holds almost everywhere,_{}_{}

(ii)_{}

(iii)_{}

[ average convergence ] if and_{}

_{}

Then the function sequence is said to be internally convergent or averagely convergent, and its limit is , denoted as_{}_{}_{}

_{}*

Average convergence has the following properties:

1 ^{o} If , then_{}_{}

_{}

established almost everywhere ._{}

2 ^{o} If , then_{}_{}

_{}

3 ^{o} If , then_{}_{}

_{}

^{}_{}The necessary and sufficient condition for the average convergence of the point sequence in 4o is that it is ^{a} basic sequence ._{}

The basic sequence is defined as follows: Let , if there is always a positive integer *N* for any, and for all such that_{}_{}_{}

_{}

is called the basic sequence in ._{}_{}

It follows that it is a complete space (see Chapter 21, § 4 , a) ._{}

[ _{}Divisibility of space ]

1 ^{o} Let , then for any , there is always a continuous function such that_{}_{}_{}

_{}

2 ^{o} Let , then for any polynomial , there is always a polynomial whose coefficients are rational numbers , so that_{}_{}_{}

_{}

Because all polynomials whose coefficients are rational numbers form a countable set and are dense everywhere in it . So 2 ^{o} is shown to be a separable space (see Chapter 21, § 3 , 3) ._{}^{}_{}

2. Hilbert space

[ Hilbert ( *H* ) space ] If each fundamental sequence in an infinite-dimensional unitary space *V* converges to an element in *V , then **V* is said to be complete . A complete infinite-dimensional unitary space is called a Hilbert space, or *H* space for short .

A vector in an *n* -dimensional space is defined as the set of n *numbers* . Similarly , a vector in an infinite-dimensional space is defined as a function of *t* going from *a* to *b* ._{}_{}

The addition and number multiplication of vectors is defined as the addition of functions and the multiplication of functions and numbers .

The formula for the inner product (quantity product) of two vectors in *H* space is

_{} ( 1 )

[ Metric of *H* space ] Let , then_{}

_{}

is the length of the vector . Let , the distance between the vector and_{}_{}_{}_{}

_{}

This expression is called the mean square error of the function and . It is to use the mean square error as a measure of the distance between two elements in the Hilbert space *H.*_{}_{}

The angle between two vectors in *H* space is defined as_{}_{}_{}

_{} ( 2 )

Because for any two functions and there are inequalities_{}_{}

_{}

So the right-hand side of equation ( 2 ) can be viewed as the cosine of an angle ._{}

[ Orthogonal function and orthogonal function system ] If the inner product of the non-zero vector *f* and *g* , it can be known from ( 1 ) and ( 2 ) , that is . Therefore, the vector *f* and *g* are said to be orthogonal . At this time_{}_{}_{}

_{}

Let represent a pairwise orthogonal function, and_{}

_{}

is their sum, then the length squared is equal to the sum of the length squares ._{}_{}

Because the length of a vector in *H* space is given by integral, it is similar to the quotient height theorem at this time given by the following formula:

_{}

The integrals described above, for example , refer to the Lebesgue integrals in a meaningful way .
_{}

If the function system in *H* space is

_{}

Any two functions in are orthogonal to each other, that is

_{}

Then this function system is called an orthogonal function system . If it also satisfies

_{}

Then this function system is called the standard orthogonal system .

[ Decomposition according to the standard orthogonal function system ] If a complete standard orthogonal function system is given in *H* space (that is, it is impossible to add a non-zero function that is orthogonal to all functions in the system), then all All functions can be expanded into series (average convergence) according to the functions in this system:_{}_{}

_{}

where the function is equal to the projection of the vector on the vector in the standard orthonormal system:_{}_{}

_{}

can prove:

_{}

Its geometric meaning is that the square of the length of a vector in *H* space is equal to the sum of the squares of the projection of the vector on the vector in the complete standard orthonormal system .

3. Banach space

[ Normalized Linear Space ] Let *V* be a linear space, for each element *α in **V* , there is a real number corresponding to it, and it has the following properties:_{}

(i) _{}, if and only then * , ;_{}_{}

(ii) _{}, in particular ;_{}

(iii) _{};

*Then V* is called a normed linear space . _{}It is called the norm or modulus of *α* .

For a normed linear space *V* ,

_{}

Then *V* becomes a scale space . Later, when we talk about normed linear space, it is always considered to be a scale space, and its distance is expressed by formula ( 1 ) .

[ Definition and Examples of Banach Spaces ] A complete normed linear space is called a Banach space .

Example 1 _{}is a Banach space .

Example 2 is assumed that the whole of all continuous functions defined within is denoted as *C* , let , belong to *C* , and *c* is any real number, define_{}_{}_{}

_{}

It is easy to know that *C* is a linear space. For *C* , the definition_{}

_{}

Then *C* is a normed linear space, which is called a space ._{}

, then the sequence of available functions converges uniformly to ._{}_{}_{}_{}

It can be shown that the space is complete, so it is a Banach space ._{}

Example 3 has a bounded real number sequence

_{}

The totality of is denoted as *M.* Let and be two bounded sequences, *a* is any real number . Define sum, number multiplication and norm as follows:_{}_{}

_{}

_{}

_{}

Then *M* becomes a normed linear space , which is called a convergent sequence space, abbreviated as space *M.* It can be proved that the space *M* is complete, so it is a Banach space .

[ Compactness ] Let *A* be a non-empty set in the scale space *E* , or any infinite subset of *A* has at least a limit point, then *A* is said to be a compact set ._{}

Every compact set must be bounded .

Let be a family of functions defined on the interval , if for any , there is always , when and when, the inequality_{}_{}_{}_{}_{}_{}

_{}

If it holds for any function in *A* , then the family of functions *A* is said to be continuous in the superior degree ._{}_{}

The Alzera-Askori theorem is assumed to be a family of continuous functions defined on , _{}_{}

like

(i) There is a constant *M* such that the functions in this family are satisfied ;_{}

(ii) *A* is continuous in the superior degree;_{}

Then there is a sequence of functions in *A* that converges uniformly above ._{}

*If A* is an element in space C, then the necessary *and sufficient conditions for **A* to be compact are: All functions in *A* are bounded and equicontinuous .

[ Linear functional and its properties ] Consider the functional *v* on the Banach space *V* , for any point *x in **V* , there is a real function corresponding to it, if_{}

(i) *v* is additive and homogeneous, that is, for any two points *x* and *y in **V* and any two real numbers *a* , *b* , there is always* *

_{}

(ii) *v* is continuous, that is, when , , then it is called a linear functional on *V.* _{}_{}_{}

Linear functionals have the following properties:

The necessary and sufficient conditions for the continuity of 1 ^{o} additive homogeneous functional are: there are constants such that_{}_{}

_{} ( 2 ) 2 ^{o} is assumed to be a linear functional, then the infimum of the set of numbers formed by all *M* satisfying ( 2 ) is called the modulus or norm, denoted as ; and there is_{}_{}_{}

_{}

3 ^{o} If a linear functional sequence { } on the Banach space *V* exists everywhere on *V* , there is a constant such that_{}_{}_{}

_{}

This is called the Uniform Bounded Principle or the Resonance Theorem .

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