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Wikipedia / Normed vector space

In mathematics, a normed vector space or normed space is a vector space, typically over the real or complex numbers, on which a norm is defined., satisfying the following four axioms:

  1. .

  2. is the zero vector.

  3. Triangle inequality

.

A norm induces a distance, called its which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but the converse is not true. For example, the set of the finite sequences of real numbers can be normed with the Euclidean norm, but it is not complete for this norm.

An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm of a Euclidean vector space is a special case that allows defining Euclidean distance

The study of normed spaces and Banach spaces is a fundamental part of functional analysis, a major subfield of mathematics.

Definition

A normed vector space is a vector space equipped with a norm. A is a vector space equipped with a seminorm.

A useful variation of the triangle inequality

This also shows that a vector norm is a (uniformly) continuous function.

), this is usually taken to be the ordinary absolute value to be the p-adic absolute value.

Topological structure

induces a metric (a notion of distance) and therefore a topology in the following sense:

  1. is jointly continuous with respect to this topology. This follows directly from the triangle inequality.

  2. is jointly continuous. This follows from the triangle inequality and homogeneity of the norm.

This turns the seminormed space into a pseudometric space (notice this is weaker than a metric) and allows the definition of notions such as continuity and convergence. To put it more abstractly every seminormed vector space is a topological vector space and thus carries a topological structure which is induced by the semi-norm.

Of special interest are complete normed spaces, which are known as and is called the

Two norms on the same vector space are called if they define the same topology. On a finite-dimensional vector space (but not infinite-dimensional vector spaces), all norms are equivalent (although the resulting metric spaces need not be the same) And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces.

is locally compact is compact is finite-dimensional; this is a consequence of Riesz's lemma. (In fact, a more general result is true: a topological vector space is locally compact if and only if it is finite-dimensional. The point here is that we don't assume the topology comes from a norm.)

The topology of a seminormed vector space has many nice properties. Given a neighbourhood system

Moreover, there exists a neighbourhood basis for the origin consisting of absorbing and convex sets. As this property is very useful in functional analysis, generalizations of normed vector spaces with this property are studied under the name locally convex spaces.

A norm (or seminorm is coarser).

Normable spaces

A topological vector space is called normable The following theorem is due to Kolmogorov:

Kolmogorov's normability criterion: A Hausdorff topological vector space is normable if and only if there exists a convex, von Neumann bounded

). that induces the quotient topology

endowed with the weak-* topology).

of the Fréchet space as defined in the article on spaces of test functions and distributions, is defined by a countable family of norms but it is

Even if a metrizable topological vector space has a topology that is defined by a family of norms, then it may nevertheless still fail to be normable space (meaning that its topology can not be defined by any norm). An example of such a space is the Fréchet space whose definition can be found in the article on spaces of test functions and distributions is defined by a countable family of norms but it is In fact, the topology of a locally convex space can be a defined by a family of if and only if there exists

Linear maps and dual spaces

The most important maps between two normed vector spaces are the continuous linear maps. Together with these maps, normed vector spaces form a category.

The norm is a continuous function on its vector space. All linear maps between finite-dimensional vector spaces are also continuous.

An isometry). Isometries are always continuous and injective. A surjective is called an isometric isomorphism are called isometrically isomorphic. Isometrically isomorphic normed vector spaces are identical for all practical purposes.

When speaking of normed vector spaces, we augment the notion of dual space is the space of all continuous is defined as the supremum into a normed vector space. An important theorem about continuous linear functionals on normed vector spaces is the Hahn–Banach theorem.

Normed spaces as quotient spaces of seminormed spaces

The definition of many normed spaces (in particular, Banach spaces) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space by the subspace of elements of seminorm zero. For instance, with the L^p spaces is a seminorm on the vector space of all functions on which the Lebesgue integral on the right hand side is defined and finite. However, the seminorm is equal to zero for any function supported on a set of Lebesgue measure zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function.

Finite product spaces

denote the product space

are norms.

this defines the same topological space.

A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm. Consequently, many of the more interesting examples and applications of seminormed spaces occur for infinite-dimensional vector spaces.

See also

References

Bibliography