There are a lot of space represented in mathematics, such as Euclidean space, normed spaces, Hilbert space. What does it matter between these spaces?
First, define the distance from the start.
What is the margin? In addition to the linear distance from the fact we often use, as well as vector distance Σni = 1xi⋅yi -------- √Σi = 1nxi⋅yi, as a function of distance ∫ba (f (x) -g (x)) 2dx∫ab (f ( x) -g (x)) 2dx, surface distance, and the like from the fold line, the relationship between the distance from the particular relationship similar to apples, bananas and other fruits, the front It is a specific thing, followed by abstract concepts. Distance is an abstract concept, which is defined as:
Let X be any non-empty set, any two points on the X x, y, there is a real number d (x, y) corresponding thereto and satisfies:
1. D (X, y) ≥≥0, and d (x, y) = 0 if and only if Y = X;
2. D (X, Y) = D (Y, X);
3. D (X, Y) ≤≤d (x, z) + d ( z, y).
He said d (x, y) is a distance in X.
After defining the distance, we add a linear structure, such as a vector addition, multiplication, addition to satisfy the commutative, associative law, zero element, the negative element; number of multiplication is commutative, a unit; the number of multiplication and addition the associativity (two) were eight requirements, so as to form a linear space, this space is a linear vector space.
In the vector space, we define the concept of norm, from a point in the space represents zero:
1. || X || ≥≥0;
2. || || AX = | A || ||| X;
3. || x + y || ≤≤ || x || + || y ||.
The norm compared with the distance, we can see, more of a norm than the distance condition 2, multiplication operations, indicating that it is a strengthening of the concept of distance. Norm distance relationship can similarly understand the relationship with the Fuji apples with apples.
Next, from the norm and extended, is formed as follows:
Norm set ⟶⟶ normed space + linear structure ⟶⟶ linear normed space
set ⟶⟶ Metric Spaces + linear structure ⟶⟶ linear distance metric space
Continue to expand in the following linear normed space has been constructed, inner product operation is added, so that the concept of angular space, is formed as follows:
Linear + normed space inner product computation ⟶⟶ inner product space;
case has an inner product space With distance, length, angle, etc., finite dimensional inner product space that is familiar Euclidean space.
Inner product space continues to expand, such that the inner product space satisfy completeness, Hilbert space is formed as follows:
an inner product space + completeness ⟶⟶ Hilbert space
which limits the completeness of the operation means that space can not run out of the space, such as a fractional rational 2-√2 space representation, its limit with the increase of the number of decimal places to converge 2-√2, but 2-√2 irrational numbers belong, are not rational in the space, it does not satisfy the complete sex. A popular understanding that the school be understood as a space, you have to go out from the dormitory began in school when school is not within the scope of the walk when to stop (limit convergence), we found that has come out of school (beyond space), the (incomplete a). Hilbert is equivalent to the Earth, no matter how you go, we are still within the Earth (except fly in space).
In addition, the aforementioned normed space to satisfy completeness, Banach space formed extended as follows:
normed space Banach space ⟶⟶ + Completeness
Above it is to add constraints on the concept of distance is formed, increasing relationship as follows:
inner product ⟶⟶ ⟶⟶ norm distance
vector space norm + + ⟶⟶ normed space linear structure ⟶ + linear structure ⟶ linear normed space + inner product calculation ⟶⟶ + completeness ⟶⟶ product space Hilbert space
inner product space ⟶⟶ + finite dimensional Euclidean space
normed space ⟶ + + completeness completeness Banach space ⟶
Incidentally following the weakening of the distance, retention limits and continuous concept of distance, to form the concept of topology
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Author: Fire greed Mito
sources : CSDN
original: https: //blog.csdn.net/shijing_0214/article/details/51052208
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