Normed vector space - Citizendia

In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rⁿ. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added It turns out that the following properties of "vector length" are the crucial ones.

The zero vector, 0, has zero length; every other vector has a positive length.
Multiplying a vector by a positive number changes its length without changing its direction. See unit vector. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length
The triangle inequality holds. In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line.

Their generalization for more abstract vector spaces, leads to the notion of norm. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length A vector space on which a norm is defined is then called a normed vector space.

1 Definition
2 Topological structure
3 Linear maps and dual spaces
4 Normed spaces as quotient spaces of semi normed spaces
5 Finite product spaces
6 See also

Definition

A semi normed vector space is a pair (V,p) where V is a vector space and p a semi norm on V. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

A normed vector space is a pair (V,||·||) where V is a vector space and ||·|| a norm on V. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

We often omit p or ||·|| and just write V for a space if it is clear from the context what (semi) norm we are using.

Topological structure

If (V, ||·||) is a normed vector space, the norm ||·|| induces a notion of distance and therefore a topology on V. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of This distance is defined in the natural way: the distance between two vectors u and v is given by ||u-v||. This topology is precisely the weakest topology that makes ||·|| continuous. Furthermore, this natural topology is compatible with the linear structure of V in the following sense:

The vector addition + : V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequality. In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater
The scalar multiplication · : K × V → V, where K is the underlying scalar field of V, is jointly continuous. This follows from the triangle inequality and homogeneity of the norm. In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater

Similarly, for any semi-normed vector space we can define the distance between two vectors u and v as ||u-v||. This turns the semi normed space into a semi metric space (notice this is weaker than a metric) and allows the definition of notions such as continuity and convergence. In Topology, a semimetric space is a generalized Metric space in which the Triangle inequality is not required In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" To put it more abstractly every semi normed vector space is a topological vector space and thus carries a topological structure which is induced by the semi-norm. In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

Of special interest are complete normed spaces called Banach spaces. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis Every normed vector space V sits as a dense subspace inside a Banach space; this Banach space is essentially uniquely defined by V and is called the completion of V. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has

All norms on a finite-dimensional vector space are equivalent from a topological point 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 V is locally compact if and only if the unit ball B = {x : ||x|| ≤ 1} is compact, which is the case if and only if V is finite-dimensional; this is a consequence of Riesz's lemma. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks Riesz's lemma is an lemma in Functional analysis. It specifies (often easy to check conditions which guarantee that a Subspace in a Normed linear (In fact, a more general result is true: a topological vector space is locally compact if and only if it is finite-dimensional. In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. The point here is that we don't assume the topology comes from a norm. )

The topology of a semi normed vector has many nice properties. Given a neighbourhood system $\mathcal{N}(0)$ around 0 we can construct all other neighbourhood systems as

$\mathcal{N}(x)= x + \mathcal{N}(0) := \{x + N \mid N \in \mathcal{N}(0) \}$

with

$x + N := \{x + n \mid n \in N \}$ . In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the

Moreover there exists a neighbourhood basis for 0 consisting of absorbing and convex sets. In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the In Functional analysis and related areas of Mathematics an absorbing set in a Vector space is a set S which can be inflated In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the 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. For functional analysis as used in psychology see the Functional analysis (psychology article In Functional analysis and related areas of Mathematics, locally convex topological vector spaces or locally convex spaces are examples of Topological

Linear maps and dual spaces

The most important maps between two normed vector spaces are the continuous linear maps. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Together with these maps, normed vector spaces form a category. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

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 f which preserves the norm (meaning ||f(v)|| = ||v|| for all vectors v). Isometries are always continuous and injective. A surjective isometry between the normed vector spaces V and W is called a isometric isomorphism, and V and W are called isometrically isomorphic. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every 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. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals The dual V ' of a normed vector space V is the space of all continuous linear maps from V to the base field (the complexes or the reals) — such linear maps are called "functionals". The norm of a functional φ is defined as the supremum of |φ(v)| where v ranges over all unit vectors (i. e. vectors of norm 1) in V. This turns V ' into a normed vector space. An important theorem about continuous linear functionals on normed vector spaces is the Hahn-Banach theorem. In Mathematics, the Hahn–Banach theorem is a central tool in Functional analysis.

Normed spaces as quotient spaces of semi normed 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. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Linear algebra, the quotient of a Vector space V by a subspace N is a vector space obtained by "collapsing" N For instance, with the L^p spaces, the function defined by

$\|f\|_p = \left( \int |f(x)|^p \;dx \right)^{1/p}$

is a seminorm on the vector space of all functions on which the Lebesgue integral on the right hand side is defined and finite. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of However, the seminorm is equal to zero for any function supported on a set of Lebesgue measure zero. In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to These functions form a subspace which we "quotient out", making them equivalent to the zero function.

Finite product spaces

Given n semi normed spaces X_i with semi norms q_i we can define the product space as

$X := \prod_{i=1}^{n} X_i$

with vector addition defined as

$(x_1,\ldots,x_n)+(y_1,\ldots,y_n):=(x_1 + y_1, \ldots x_n + y_n)$

and scalar multiplication defined as

$\alpha(x_1,\ldots,x_n):=(\alpha x_1, \ldots, \alpha x_n)$ . In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural

We define a new function q

$q:X \mapsto \mathbb{R}$

for example as

$q:(x_1,\ldots,x_n) \to \sum_{i=1}^n q_i(x_i)$ .

which is a seminorm on X. The function q is a norm if and only if all q_i are norms.

More generally, for each real p≥1 we have the seminorm:

$q:(x_1,\ldots,x_n) \to \left( \sum_{i=1}^n q_i(x_i)^p \right)^\frac{1}{p}$

For each p 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.

Contents

Definition

Topological structure

Linear maps and dual spaces

Normed spaces as quotient spaces of semi normed spaces

Finite product spaces

See also