1. What is linear space
1. Linear space
Definition: Let V be a non-empty set and F be a number field. If for any two elements α, β∈V, there is always a unique element γ∈V corresponding to it, which is called the sum of α and β, denoted as
γ = α + β
If for any number λ∈F and any element α∈V, there is always a unique element δ∈V corresponding to it, which is called the quantity product of λ and α, denoted as
δ = λα
If the above two operations satisfy the following eight operation rules, then V is called a linear space on the number field F
Addition operation:
① Commutative law: α+β = β+α
② 结合 律 ： (α + β） + γ = α + （β + γ）
③ Zero element (unique): there is 0∈V, for any α∈V, let α+0=α
④ Negative element (unique): for any α∈V, there is β∈V, so that α+β=0
Multiplication operation:
⑤ 1α = α
⑥ (λμ) α = λ (μα)
⑦ (λ + μ) α = λα + μα
⑧ λ (α + β) = λα + λβ
2. Number field
Definition: Let F be the set of numbers that satisfy
① 0，1∈F
② For any two numbers a, b in F, there is always a+b, ab, a×b, a÷b(b≠0)∈F
Then F is said to be a number field, and F is said to be closed to the four operations of addition, subtraction, multiplication and division.
3. Properties of Linear Spaces
① Zero element is unique;
② Negative elements are unique;
③ If λα = 0 (λ∈F,α∈V), then k=0 or λ=0
④ (-λ) α = λ (-α) = - (λα)
4. Summary
A linear space can be formed only if the following conditions are met
① The number set F is a number field;
② The set V satisfies the eight laws of addition and multiplication
2. From distance to Hilbert space
1. What is distance
In fact, in addition to the straight-line distance we often use, there are also vector distances 
, function distances such as 
surface distances, polyline distances, etc. The relationship between these specific distances and distances is similar to the relationship between apples, bananas, etc. and fruits , the front is the concrete thing, the back is the smoke
Elephant Concept. Distance is an abstract concept defined as :
Let X be any non-empty set, for any two points x, y in X, there is a real number d(x, y) corresponding to it and satisfy:
1. d(x, y) ≥ ≥ 0, and d( x, y) y)=0 if and only if x=y;
2. d(x,y)=d(y,x);
3. d(x,y) ≤ ≤d(x,z)+d(z,y ).
Call d(x, y) a distance (metric) in X, and call X a metric space for metric d.
After defining the distance, we add linear structures , such as vector addition and number multiplication, to make it satisfy the commutative law, associative law, zero element and negative element of addition; commutative law of number multiplication, unit one; number multiplication and addition The associativity of (two) has a total of eight requirements, thus forming a linear space, and this linear space is a vector space .
In vector space, we define the concept of norm, which represents the distance from a point to the zero point of space:
1. ||x|| ≥ ≥0;
2. ||ax||=|a|||x||;
3. ||x+y|| ≤ ≤||x||+||y||.
Comparing the norm with the distance, it can be seen that the norm has one more condition 2 than the distance, and the operation of multiplying numbers indicates that it is an enhanced concept of distance. The relationship between norm and distance can be similarly understood as the relationship between red Fuji apples and apples.
distance + linear space = linear metric space
Normal Form + Linear Space = (Linear) Normed Space
Linear normed space + inner product operation = inner product space
Linear normed space + completeness = Banach space
At this time, the inner product space already has distance, length, angle, etc. The finite-dimensional inner product space is also the familiar Euclidean space.
Inner product space + finite dimension = Euclidean space
Inner product space + completeness = Hilbert space