In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that is commonly denoted and has the following properties:
which make any normed vector space into a metric space and a topological vector space. If this metric 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 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.
Property 3 depends on a choice of norm on the field of scalars. When the scalar field is (or more generally a subset of ), this is usually taken to be the ordinary absolute value, but other choices are possible. For example, for a vector space over one could take to be the -adic norm.
If is a normed vector space, the norm 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 and is given by This topology is precisely the weakest topology which makes continuous and which is compatible with the linear structure of in the following sense:
The vector addition is jointly continuous with respect to this topology. This follows directly from the triangle inequality.
The scalar multiplication where is the underlying scalar field of is jointly continuous. This follows from the triangle inequality and homogeneity of the norm.
Similarly, for any seminormed vector space we can define the distance between two vectors and as 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 Banach spaces.
Every normed vector space sits as a dense subspace inside some Banach space; this Banach space is essentially uniquely defined by and is called the completion of
Two norms on the same vector space are called equivalent if they define the same topology. On a finite-dimensional vector space, all norms are equivalent but this is not true for infinite dimensional vector spaces.
All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (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. A normed vector space is locally compact if and only if the unit ball is compact, which is the case if and only if 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 around 0 we can construct all other neighbourhood systems as
A norm (or seminorm) on a topological vector space is continuous if and only if the topology that induces on is coarser than (meaning, ), which happens if and only if there exists some open ball in (such as maybe for example) that is open in (said different, such that ).
A product of a family of normable spaces is normable if and only if only finitely many of the spaces are non-trivial (that is, ). Furthermore, the quotient of a normable space by a closed vector subspace is normable, and if in addition 's topology is given by a norm then the map given by is a well defined norm on that induces the quotient topology on 
Furthermore, is finite dimensional if and only if is normable (here denotes endowed with the weak-* topology).
The 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 not a normable space because there does not exist any norm on such that the topology that this norm induces is equal to
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).
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, because its topology is defined by a countable family of norms but it is not a normable space because there does not exist any norm on such that the topology this norm induces is equal to
In fact, the topology of a locally convex space can be a defined by a family of norms on if and only if there exists at least one continuous norm on 
Linear maps and dual spaces
The most important maps between two normed vector spaces are the continuouslinear 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 between two normed vector spaces is a linear map which preserves the norm (meaning for all vectors ). Isometries are always continuous and injective. A surjective isometry between the normed vector spaces and is called an isometric isomorphism, and and 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 to take the norm into account. The dual of a normed vector space is the space of all continuous linear maps from to the base field (the complexes or the reals) — such linear maps are called "functionals". The norm of a functional is defined as the supremum of where ranges over all unit vectors (that is, vectors of norm ) in This turns 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 spaces, the function defined by
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
Given seminormed spaces with seminorms denote the product space by
where vector addition defined as
and scalar multiplication defined as
Define a new function by
which is a seminorm on The function is a norm if and only if all are norms.
More generally, for each real the map defined by
is a semi norm.
For each 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.
Banach space, normed vector spaces which are complete with respect to the metric induced by the norm
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Schaefer, H. H. (1999). Topological Vector Spaces. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Trèves, François (2006) . Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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