Normed vector space

A normed space or normed vector space is in mathematics a vector space on which a norm is defined. Every normed space is induced by the standard metric is a metric space with the metric induced by this topology is a topological space. If a normed space completely, so it is called a complete normed space or Banach space. A normed space can be derived from a pre-Hilbert space on Skalarproduktnorm or of a vector space with half- standard as space factor.

Normed spaces are a central object of study in functional analysis and play an important role in the solution structure of partial differential equations and integral equations.

• 5.1 Skalarprodukträume
• 5.2 Semi- Normed spaces

Definition

If a vector space over the field of real or complex numbers and a norm on, then called the couple a normed vector space. A norm is a mapping which assigns a non-negative real number to an element of the vector space and has the three properties definiteness, absolute homogeneity and subadditivity. That is a standard, if all is true of all of the vector space and:

If it is clear to which standard it is, you can also do without their explicit specification and then writes only for the normalized space.

History

Hermann Minkowski used from 1896 to investigate number-theoretic questions after today's terminology finite dimensional normed vector spaces. The axiomatic definition of the vector space asserted itself only in the 1920s. Minkowski found that it is only necessary to define a distance compatible with the vector structure, specify the calibration body. A calibration table is the set of all vectors with the norm or length less than or equal to one. For example, the full sphere with radius one a calibration body. Minkowski also found that the calibration body is a convex and respect to the coordinate origin centrally symmetric subset, see Minkowski functional.

The now common standard symbol was first used by Erhard Schmidt 1908. His work laid it close to the expression as the distance between the vectors and interpreted. In a published work Frigyes Riesz in 1918 used the standard systematic symbol for the supremum norm on the space of continuous functions on a compact interval.

After preliminary work by Henri Lebesgue in the years 1910 and 1913 Stefan Banach developed in his dissertation of 1922, the axiomatic definition of the Standard, and the normalized vector space. According to him, the complete normed vector spaces, the Banach spaces named.

Examples

The following normed spaces are all also complete:

• The space of real or complex numbers with the sum norm:
• The space of real or complex vectors with the p-norm:
• The space of real or complex matrices with the Frobenius norm:
• The space of real - or complex-valued summable in p- th power consequences with the ℓ p-norm:
• The space of real - or complex-valued bounded functions with the supremum norm:
• The space of real - or complex-valued continuous functions on a compact set of definitions with the maximum norm:
• The space of real - or complex-valued in the p- th power Lebesgue integrable functions with the Lp norm:
• The space of real - or complex- restricted m-times continuously differentiable functions with the Cm - norm:

The following example is if and only complete if the vector space is complete:

• The space of bounded linear operators between two real or complex vector space with the operator norm:

Properties

Completeness

A normed space is called complete if every Cauchy sequence in this space has a limit. A complete normed space is called a Banach space. Every normed space can be completed by the formation of equivalence classes of Cauchy sequences. In this way one obtains a Banach space that contains the original space as a dense subspace.

Metric and topology

Every norm induced by

A metric. Every normed space is thus also be a metric space and continue with the norm topology, a topological space. Thus, topological concepts as limit, Cauchy sequence, continuity and compactness are defined in normed spaces. Thus, a sequence converges if and only to a limit if and only if. The standard itself is a continuous map with respect to the topology induced by it.

Equivalent norms induce the same uniform structure and thus the same topology. In finite-dimensional vector space all norms are equivalent in infinite-dimensional spaces but this is not the case.

Related Facilities

Skalarprodukträume

A standard may, but need not, by a scalar product (inner product ) must be defined. Every inner product space with the scalar product induced by the norm

A normed space. A norm is induced by an inner product if and only if the parallelogram law is fulfilled in the resulting space.

A complete inner product space is called a Hilbert space.

Half Normed spaces

If only one semi-norm, it is called a semi- normed space. For a room with a semi-norm normed space obtained as the factor space. These elements and each other are identified that meet. In the functional analysis is considered in addition to the normed spaces and vector spaces with a lot of semi-norms and thus comes to the concept of a locally convex space.