Let’s first talk about what isHilbert space。
This concept sounds tall, but it’s actually a very simple concept.
First of all, what is itlinear space
Linear space is the space that defines multiplication and addition. This is a space with a linear structure. With the concept of linear space, we can find a set of bases in the space because of multiplication and addition. We can get all the points in the space by linear combination.
Metric space and normed space
The definition of distance must satisfy the following three conditions:
- d(x,y)≥0; D (x, y) = 0 if and only if x = y, that is, nonnegativity
- d(x,y)=d(y,x); Symmetry
- D (x, z) + D (Z, y) ≥ D (x, y) satisfies trigonometric inequality.
The space that defines the distance is calledmetric space。
A linear space with distance defined is calledLinear metric space
Next, we define the norm ||||||||||||||||||||||||
- ||X | ≥ 0 is nonnegative
- ||X | + |, y | ≥ |, x + y |, satisfies trigonometric inequality
So the concept of norm can be seen as the distance from zero to x, plus the second|| α x||=| α||| X |, that is, number multiplication can be extracted.
**Distance can be defined by norm, i.e. D (x, y) = | – x − y |, but distance cannot be defined by norm
Because the definition of distance does not satisfy the second condition of norm, norm is more specific than distance (fruit and tropical fruit)**
The space of norm defined is called normed space and metric space. in additioncompleteThe normed space of Banach space is called Banach space.
And the linear space that defines norm is calledNormed linear space