Norm is a function with the concept of "length". In the fields of linear algebra, functional analysis and related mathematics, norm function is a function that gives all vectors in vector space a non-zero positive length or size. On the contrary, a semi-norm can give a nonzero vector a zero length.

For a simple example, Euclid norm can be defined in two-dimensional Euclid geometric space R, and the elements in this vector space are often drawn as a directed line segment with an arrow from the origin in Cartesian coordinate system. The Euclidean norm of each vector is the length of the directed line segment.

The vector space that defines the norm is a normed vector space. Similarly, a semi-norm vector space is defined as a normed vector space.

Suppose v is a vector space over field f; The seminorm of v is a function; , content with:

(non-negative)

(positive homogeneity)

Trigonometric inequality

Norm is semi-norm plus additional properties:

P(v) is a zero vector. If and only if v is a zero vector (positive definiteness), if the topology of a topological vector space can be derived from norms, then this topological vector space is called a normed vector space.

If x is a linear space on the number field k, then the functional satisfies:

1. Positive definiteness:, and;

4. Positive homogeneity:

3. Subadditivity (trigonometric inequality):.

Then call it a norm on x.

(Note that if ║x+y║≤║x║+║y║, reuse ║-x = ║x ║ to get ║ x).

If a norm is defined in a linear space, it is called a normed linear space.

Note: Norm is related to inner product, metric and topology.

1. metric: it can be derived from the norm, and then the topology can be derived, so the normed linear space is the metric space.

But on the other hand, measurement is not necessarily induced by norms.

If a normed linear space is complete as a metric space (its norm naturally induces the metric d(x,y)=║x-y║), that is, any Cauchy sequence converges in it, then this normed linear space is called a Banach space.

3. Use inner product; Can sum up the norms:.

Conversely, norm does not necessarily induce inner product. When the norm satisfies the parallelogram formula, this norm can definitely induce the inner product.

A complete inner product space is called Hilbert space.

4. If the positive definiteness in the norm definition is removed, the obtained functional is called semi-norm or quasi-norm, and the corresponding linear space is called quasi-norm linear space.

For the two norms on x, if there is a normal number c that satisfies

Then say weaker than. If weaker than and weaker than, the two norms are said to be equivalent.

It can be proved that the norms in finite dimensional space are all equivalent, and there is at least Alef (radix of real number set) unequal norm in infinite dimensional space.