Normed vector space

[1] A norm is a generalization of the intuitive notion of "length" in the physical world.

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

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

induces a metric (a notion of distance) and therefore a topology on

This metric is defined in the natural way: the distance between two vectors

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.

sits as a dense subspace inside some Banach space; this Banach space is essentially uniquely defined by

Two norms on the same vector space are called equivalent if they define the same topology.

(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.

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.

The following theorem is due to Kolmogorov:[3] Kolmogorov's normability criterion: A Hausdorff topological vector space is normable if and only if there exists a convex, von Neumann bounded neighborhood of

is a Hausdorff locally convex topological vector space then the following are equivalent: Furthermore,

as defined in the article on spaces of test functions and distributions, is defined by a countable family of norms but it is not a normable space because there does not exist any norm

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 single norm).

whose definition can be found in the article on spaces of test functions and distributions, because its topology

In fact, the topology of a locally convex space

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 between two normed vector spaces is a linear map

A surjective isometry between the normed vector spaces

Isometrically isomorphic normed vector spaces are identical for all practical purposes.

to the base field (the complexes or the reals) — such linear maps are called "functionals".

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.