Normed vector space

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Hierarchy of mathematical spaces. Normed vector spaces are a superset of inner product spaces and a subset of metric spaces, which in turn is a subset of topological spaces. Mathematical Spaces.png
Hierarchy of mathematical spaces. Normed vector spaces are a superset of inner product spaces and a subset of metric spaces, which in turn is a subset of topological spaces.

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. [1] 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: [2]

Contents

  1. It is nonnegative, that is for every vector x, one has
  2. It is positive on nonzero vectors, that is,
  3. For every vector x, and every scalar one has
  4. The triangle inequality holds; that is, for every vectors x and y, one has

A norm induces a distance, called its (norm) induced metric , by the formula

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.

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 by the formula

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

Definition

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

A useful variation of the triangle inequality is

for any vectors x and y.

This also shows that a vector norm is a continuous function.

Property 2 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 p-adic norm.

Topological structure

If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric (a notion of distance) and therefore a topology on V. This metric 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 which makes ‖·‖ continuous and which is compatible with the linear structure of V in the following sense:

  1. The vector addition + : V × VV is jointly continuous with respect to this topology. This follows directly from the triangle inequality.
  2. 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.

Similarly, for any semi-normed vector space we can define the distance between two vectors u and v as ‖u  v‖. 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 semi-normed 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 called Banach spaces. 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.

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). [3] 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 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

with

.

Moreover, there exists a neighbourhood basis for 0 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.

Normable spaces

A topological vector space is called normable if there exists a norm on X such that the canonical metric induces the topology on X. The following theorem is due to Kolmogorov: [4]

Theorem A Hausdorff topological vector space is normable if and only if there exists a convex, von Neumann bounded neighborhood of .

A product of a family of normable spaces is normable if and only if only finitely many of the spaces are non-trivial (i.e. ). [4] Furthermore, the quotient of a normable space X by a closed vector subspace C is normable and if in addition X's topology is given by a norm then the map given by is a well defined norm on X/C that induces the quotient topology on X/C. [5]

If X is a Hausdorff locally convex topological vector space then the following are equivalent:

  1. X is normable.
  2. X has a bounded neighborhood of the origin.
  3. the strong dual of X is normable. [6]
  4. the strong dual of X is metrizable. [6]

Furthermore, X is finite dimensional if and only if is normable (here denotes endowed with the weak-* topology).

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 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 an isometric isomorphism, and V and W 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 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.

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 Lp 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 n seminormed spaces Xi with seminorms qi we can define the product space as

with vector addition defined as

and scalar multiplication defined as

.

We define a new function q

for example as

.

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

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

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.

See also

Related Research Articles

In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, any vector space has a corresponding dual vector space consisting of all linear forms on , together with the vector space structure of pointwise addition and scalar multiplication by constants.

The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.

Vector space Basic algebraic structure of linear algebra

A vector space is a set of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. To specify that the scalars are real or complex numbers, the terms real vector space and complex vector space are often used.

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In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × V → ℝ so that

  1. Scalar multiplication in V is continuous with respect to d and the standard metric on ℝ or ℂ.
  2. Addition in V is continuous with respect to d.
  3. The metric is translation-invariant; i.e., d(x + a, y + a) = d(x, y) for all x, y and a in V
  4. The metric space (V, d) is complete.

In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) such that the canonical evaluation map from X into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. Note that in 1951, R. C. James discovered a non-reflexive Banach space that is isometrically isomorphic to its bidual.

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.

In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In mathematics, a norm is a function from a real or complex vector space to the nonnegative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.

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In mathematics, a real coordinate space of dimension n, written Rn or , is a coordinate space over the real numbers. This means that it is the set of the n-tuples of real numbers. With component-wise addition and scalar multiplication, it is a real vector space.

In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.

In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products, but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.

The strongest locally convex topological vector space (TVS) topology on , the tensor product of two locally convex TVSs, making the canonical map continuous is called the projective topology or the π-topology. When X ⊗ Y is endowed with this topology then it is denoted by and called the projective tensor product of X and Y.

This is a glossary for the terminology in a mathematical field of functional analysis.

In functional analysis and related areas of mathematics, a metrizable topological vector spaces (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

References

  1. Callier, Frank M. (1991). Linear System Theory. New York: Springer-Verlag. ISBN   0-387-97573-X.
  2. Rudin 1991, pp. 3-4.
  3. Kedlaya, Kiran S. (2010), p-adic differential equations, Cambridge Studies in Advanced Mathematics, 125, Cambridge University Press, CiteSeerX   10.1.1.165.270 , ISBN   978-0-521-76879-5 , Theorem 1.3.6
  4. 1 2 Schaefer 1999, p. 41.
  5. Schaefer 1999, p. 42.
  6. 1 2 Trèves 2006, pp. 136–149, 195–201, 240–252, 335–390, 420–433.

Bibliography