Before that, the space we are concerned about is basically a space composed of function space or sequence, and the concepts of distance space, normed space, inner product space and Hilbert space are all established.
Using mathematical research methods such as analogy, association and induction, the algebraic structure and geometric characteristics of finite-dimensional space are extended to infinite-dimensional space.
Many mathematical problems, such as translation and rotation in analytic geometry in middle school, are just linear transformations (operations).
Differential and integral in advanced mathematics are also linear operations, which have many similar operational properties (rotation, stretching and translation of vectors, etc.) to linear transformation in space. ).
Linear equations, differential equations and integral equations can all be regarded as linear operations (or linear transformations or linear mappings) in a specific space.
We call these linear operators, which is one of the most important basic concepts in functional analysis. We put all bounded linear operators (such as integrals, matrices, etc. ) as a linear space, and give it a norm to become a normed linear space, and linear operators are regarded as elements in the normed space.
Linear operator space is the main object of linear functional analysis. In the framework of linear operator space, the properties of linear operations are studied to solve problems in analysis, algebra and geometry.
By discussing the essential characteristics of bounded linear operators in normed spaces, we can get some profound conclusions:
Operations that satisfy properties are called linear operators. So differential operation and integral operation are linear operators.
Definition 1: Let it be a normed space, a linear subspace and a mapping from to, satisfying:
Where (is a numeric field), the mapping is called a linear operator from to. A domain named.
Note 1: Generally, it is a proper subset. If it is, it is called a top-down linear operator.
Note 2: If (number),. Such a linear operator is called a linear functional.
That is, linear functional (or) is a linear operator from normed space to number domain.
Note 3: From the point of view of signal and system, space is actually the input space (input signal) of the system, and space is the output space (output signal) of the system or the space after transformation (action); Linear operators are linear systems.
Definition 2: Let it be a linear operator from to. If there is a constant, it makes
It is called bounded linear operator.
If a linear functional is bounded and there is a constant, then
It is called bounded functional. Because the functional is mapped into a number, the norm of the number is represented by an absolute value.
Note 1: Boundedness of bounded linear operators means that the "magnification" after mapping does not exceed a constant. (The size of an element is measured by a norm)
Note 2: Because the inner product can produce norms and the inner product space is also a normed space, the discussion about bounded linear operators and bounded functionals in normed space is still valid in the inner product space.
Note 3: Bounded linear operators map bounded sets into bounded sets (bounded inputs, bounded outputs).
Definition 4: Let it be a normed space and a linear operator from to. If so, it is said to be continuous at that point.
Theorem 5: Let it be a normed space and a linear operator from to. If it is continuous at this point, it is continuous at this point.
Note 1: For linear operators, a little continuity is a little continuity.
Note 2: The continuity of linear operators means:
Limit operation and linear operation can be interchanged in order.
Theorem 6: Let a normed space and a linear operator from to be continuous if and only if it is bounded.
Next, we regard the bounded linear operator as an element and form a new linear space, that is, a space composed of all bounded linear operators (such as integral operation and matrix operation).
The properties of linear operators are studied from the perspective of normed space.
Definition 7: Let it be a normed space, representing all bounded linear operators from to.
If, let's remember Jane.
Linear operations (addition, digital multiplication) can be naturally performed in. For any sum, define:
Because addition and multiplication operations are closed, they become linear spaces.
Next, we regard bounded linear operators as elements in space and define the norm of bounded linear operators in space.
Definition 8: Let it be a bounded linear operator from normed space to, that is, it exists so that
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It's called the norm of linear operators.
Theorem 10: Let it be a bounded linear operator of normed space, then:
In particular, at that time, you can also define multiplication operations (expressed as):
Obviously also a linear operator, and:
In addition, there are:
Theorem 1 1: Hypothesis. For any, please define:
It is a bounded functional on.
Note: any bounded functional on a can be written in the above form, that is, the bounded functional on a can be determined by the elements in.
The bounded functional on infinite dimensional space is given below.
Theorem 12: Let be a continuous function. For any, define:
It is a bounded functional on.
Note 1: The norm of linear functional can be proved.
Note 2: In particular, if the definite integral is a bounded functional on.
Note 3: Not all linear operators are bounded. For example, a very important differential operator is an unbounded operator. For example, to distinguish, we have:
However, it is unbounded.
Note: Differential operator is a very important unbounded linear operator. Although the differential operator is unbounded, it is a closed linear operator. Closed linear operators also have the good property of "quasi-continuity"
The operator's distance can be induced by the operator's specifications:
Therefore, it is also a distance space (the distance structure of elements is defined in the space). With the distance, we can discuss the convergence of the sequence of elements in the space, and then we can discuss the completeness of the space.
Obviously, the problem of convergence of operator sequences in norm can be solved in.
Defines the 1: setting, if
Then the sequence of bounded linear operators converges to bounded linear operators according to the norm.
Theorem 2: The norm convergence of linear operator sequence in space is equivalent to the uniform convergence of linear operator sequence on unit sphere (the convergence speed is independent of the value).
Uniform convergence is an intuitive explanation. Let all the largest points converge, then other points will inevitably converge, which is determined by the definition of operator norm, which takes the maximum magnification (operators have different magnifications for different values).
In addition, the convergence of operator sequences under norm is equivalent to uniform convergence on bounded sets.
Linear operators can define other convergence modes besides norm convergence (or uniform convergence) in space.
Definition 3: Settings. If yes, that is
It is called point-by-point convergence to (different convergence rates may be different) or strong convergence to.
Note: Normalized convergence to (uniform convergence) can lead to strong convergence to, and vice versa.
The space composed of bounded linear operators is normed, so its completeness can be discussed.
A normed space is complete if and only if the columns in the space converge.
Theorem 5: Let it be a normed space and a space, then the bounded linear operator space is a space (complete normed space).
We abstract linear operators into elements in the space of linear operators. The purpose of abstraction is to let us see some essential characteristics of linear operators more clearly.
In the framework of linear operator space, studying the properties of linear operations will lead to some profound conclusions, such as uniform boundedness principle, open mapping theorem, inverse operator theorem and closed image theorem. These three theorems and theorems (extension theorems of linear functional) can be regarded as the cornerstones of linear operator theory in normed space.
These three theorems characterize the important properties of linear operators in space.
Definition 1: Let it be a distance space. If any non-empty open set is dense, it is called sparse set.
Definition of Dense: It is a set of points in distance space, and if it is, it is called medium dense.
Note: There is no interior point in sparse set. In fact, if it is an inside point, there is a kick-off, so it is dense in the kick-off.
Definition 2: If a set can be represented as the union of at most a few sparse sets, that is
Among them, the sparse set is called the first contour set. A set that is not the first outline set is called the second outline set.
Theorem 3 (Contour Theorem): A complete distance space is the second contour set.
Inference: Space is a set of the second kind.
For bounded linear operators, we can get that a family of bounded linear operators with bounded points must be uniformly bounded.
Theorem 7 (uniformly bounded principle):
Let it be a family of bounded linear operators from space to normed space. If so, yes.
It is a bounded set. Belongs to an indicator set.
Note 1: "Consistency" means that it is true for all.
Note 2: This theorem shows that if there is any existence that makes
Have a common, so
The negative proposition of this theorem: if it is a family of bounded linear operators from space to normed space, then it exists, so
This proposition is called the resonance theorem.
According to Theorem 5 in the previous section, if it is a normed space or space, then a bounded linear operator space is a space. That is, any column in the space converges according to the norm of the operator (that is, when, the operator norm).
First, consider the completeness in the sense of strong convergence, that is, any column in the space converges point by point.
Theorem 12: Let a space be complete in the sense of strong convergence.
Note: The meaning of integrity:
Note: It is the output of the operator (model) after the first iteration. .
If there is only uniqueness for any given mapping (representing the value range of the mapping), the mapping is said to be injective. At this point, you can define an operator from the range to, which is called the inverse operator of.
Definition 1 (inverse operator): Let it be a linear operator from linear space to linear space. If there is a linear operator in, then
Then the operator is said to have an inverse operator, and the inverse operator of is denoted as.
Note 1: The necessary and sufficient condition for the existence of the inverse operator is that it is a one-to-one mapping from space to space.
Note 2: If it exists, it is unique.
Note 3: It can be proved that it is also a linear operator.
Note 4:.
Theorem 2: Let it be a linear operator from normed space to normed space. If it exists, then
There is a bounded inverse operator.
Note 1: It is a mapping from to, not necessarily the whole space, and not necessarily the whole space.
Note 2: Boundedness is not required here, just below.
Definition 3: Let it be a mapping. If any open set in is mapped to an open set in, it is called an open mapping.
Theorem 4 (open mapping theorem): Let space be defined.