### Hilbert space

Let’s first talk about what is**Hilbert space**。

This concept sounds tall, but it’s actually a very simple concept.

First of all, what is it**linear space**

### linear 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 called**metric space**。

A linear space with distance defined is called**Linear metric space**

Next, we define the norm ||||||||||||||||||||||||

- ||X | ≥ 0 is nonnegative
- ||αx||=|α|||x||
- ||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.

So:

**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 addition**complete**The normed space of Banach space is called Banach space.

And the linear space that defines norm is called**Normed linear space**