An
Abelian group
,
written additively, in which a multiplication of the
elements by scalars is defined, i.e. a mapping
which satisfies the following axioms
(
;
):
1)
;
2)
;
3)
;
4)
.
Axioms 1)–4) imply the following important
properties of a vector space
(
):
5)
;
6)
;
7)
.
The elements of the vector space are called its
points,
or
vectors;
the elements of
are called
scalars.
The vector spaces most often employed in mathematics and
in its applications are those over the field
of complex numbers and over the field
of real numbers; they are said to be
complex,
respectively
real,
vector spaces.
The axioms of vector spaces express algebraic properties of
many classes of objects which are frequently encountered in analysis.
The most fundamental and the earliest examples of vector spaces are the
-dimensional
Euclidean spaces. Of almost equal importance are many function
spaces: spaces of continuous functions, spaces of measurable
functions, spaces of summable functions, spaces of analytic
functions, and spaces of functions of bounded variation.
The concept of a vector space is a special case of the concept
of a module over a ring — a vector space is a
unitary module
over a field. A unitary module over a non-commutative skew-field is also called a
vector space over a skew-field;
the theory of such vector spaces is much more difficult
than the theory of vector spaces over a field.
One important task connected with vector spaces is the study of the
geometry of vector spaces, i.e. the study of lines
in vector spaces, flat and convex sets in vector
spaces, vector subspaces, and bases in vector spaces.
A
vector subspace,
or simply a
subspace,
of a vector space
is a subset
that is closed with respect to the operations of addition and multiplication
by a scalar. A subspace, considered apart from its ambient
space, is a vector space over the ground field.
The
straight line
passing through two points
and
of a vector space
is the set of elements
of the form
,
.
A set
is said to be a
flat set
if, in addition to two arbitrary points, it also
contains the straight line passing through these points. Any
flat set is obtained from some subspace by a parallel shift:
;
this means that each element
can be uniquely represented in the form
,
,
and that this equation realizes a one-to-one correspondence between
and
.
The totality of all shifts
of a given subspace
forms a vector space over
,
called the
quotient space
,
if the operations are defined as follows:
Let
be an arbitrary set of vectors in
.
A
linear combination of the vectors
is a vector
defined by an expression
in which only a finite number of coefficients differ from zero.
The set of all linear combinations of vectors of the set
is the smallest subspace containing
and is said to be the
linear envelope of the set
.
A linear combination is said to be
trivial
if all coefficients
are zero. The set
is said to be a
linearly independent set
if all non-trivial linear combinations of vectors in
are non-zero.
Any linearly independent set is contained in some maximal linearly independent set
,
i.e. in a set which ceases to be linearly independent after any element in
has been added to it.
Each element
may be uniquely represented as a linear combination
of elements of a maximal linearly independent set:
A maximal linearly independent set is said to be a
basis
(an
algebraic basis)
of the vector space for this reason. All bases
of a given vector space have the same cardinality, which is known as the
dimension of the vector space.
If this cardinality is finite, the space is said to be
finite-dimensional;
otherwise it is known as an
infinite-dimensional vector space.
The field
may be considered as a one-dimensional vector space over itself;
a basis of this vector space is a single element, which may be
any element other than zero. A finite-dimensional vector space with a basis of
elements is known as an
-dimensional space.
The theory of convex sets plays an important part in
the theory of real and complex vector spaces (cf. also
Convex set).
A set
in a real vector space is said to be a
convex set
if for any two points
in it the segment
,
,
also belongs to
.
The theory of linear functionals on vector spaces and the
related theory of duality are important parts of the theory of vector spaces. Let
be a vector space over a field
.
An additive and homogeneous mapping
,
i.e.
is said to be a
linear functional
on
.
The set
of all linear functionals on
forms a vector space over
with respect to the operations
This vector space is said to be the
conjugate,
or
dual,
space
of
.
Several geometrical notions are connected with the concept of a conjugate space. Let
(respectively,
);
the set
or
,
is said to be the
annihilator
or
orthogonal complement
of
(respectively, of
);
here
and
are subspaces of
and
,
respectively. If
is a non-zero element of
,
is a maximal proper linear subspace in
,
which is sometimes called a
hypersubspace;
a shift of such a subspace is said to be a
hyperplane
in
;
thus, any hyperplane has the form
If
is a subspace of the vector space
,
there exist natural isomorphisms between
and
and between
and
.
A subset
is said to be a
total subset
over
if its annihilator contains only the zero element,
.
Each linearly independent set
can be brought into correspondence with a
conjugate set
,
i.e. with a set such that
(the
Kronecker symbol)
for all
.
The set of pairs
is said to be a
biorthogonal system.
If the set
is a basis in
,
then
is total over
.
An important chapter in the theory of vector spaces is the theory of
linear transformations
of these spaces. Let
be two vector spaces over the same field
.
Then an additive and homogeneous mapping
of
into
,
i.e.
is said to be a
linear mapping
or
linear operator,
mapping
into
(or from
into
).
A special case of this concept is a
linear functional,
or a linear operator from
into
.
An example of a linear mapping is the natural mapping from
into the quotient space
,
which establishes a one-to-one correspondence between each element
and the flat set
.
The set
of all linear operators
forms a vector space with respect to the operations
Two vector spaces
and
are said to be
isomorphic
if there exists a linear operator (an
"isomorphism" )
which realizes a one-to-one correspondence between their elements.
and
are isomorphic if and only if their bases have equal cardinalities.
Let
be a linear operator from
into
.
The
conjugate linear operator,
or
dual linear operator,
of
is the linear operator
from
into
defined by the equation
The relations
,
are valid, which imply that
is an isomorphism if and only if
is an isomorphism.
The theory of bilinear and multilinear mappings of
vector spaces is closely connected with the theory
of linear mappings of vector spaces (cf.
Bilinear mapping;
Multilinear mapping).
Problems of extending linear mappings are an important group
of problems in the theory of vector spaces. Let
be a subspace of a vector space
,
let
be a linear space over the same field as
and let
be a linear mapping from
into
;
it is required to find an extension
of
which is defined on all of
and which is a linear mapping from
into
.
Such an extension always exists, but the problem may
prove to be unsolvable owing to additional limitations imposed
on the functions (which are related to supplementary structures in the
vector space, e.g. to the topology or to an
order relation). Examples of solutions of extension problems are the
Hahn–Banach theorem
and theorems on the extension of positive functionals in spaces with a cone.
An important branch of the theory of vector spaces is the
theory of operations over a vector space, i.e. methods for
constructing new vector spaces from given vector spaces. Examples of such
operations are the well-known methods of taking a subspace and
forming the quotient space by it. Other important
operations include the construction of direct sums, direct
products and tensor products of vector spaces.
Let
be a family of vector spaces over a field
.
The set
which is the product of
can be made into a vector space over
by introducing the operations
The resulting vector space
is called the
direct product
of the vector spaces
,
and is written as
.
The subspace of the vector space
consisting of all sequences
for each of which the set
is finite, is said to be the
direct sum
of the vector spaces
,
and is written as
or
.
These two notions coincide if the number of terms
is finite. In this case one uses the notations:
or
Let
and
be vector spaces over the same field
;
let
,
be total subspaces of the vector spaces
,
,
and let
be the vector space with the set of all elements of the space
as its basis. Each element
can be brought into correspondence with a bilinear function
on
using the formula
,
,
.
This mapping on the basis vectors
may be extended to a linear mapping
from the vector space
into the vector space of all bilinear functionals on
.
Let
.
The
tensor product
of
and
is the quotient space
;
the image of the element
is written as
.
The vector space
is isomorphic to the vector space of bilinear functionals on
(cf.
Tensor product
of vector spaces).
The most interesting part of the theory of vector spaces
is the theory of finite-dimensional vector spaces. However, the
concept of infinite-dimensional vector spaces has also proved
fruitful and has interesting applications, especially in the theory of
topological vector spaces,
i.e. vector spaces equipped with topologies fitted
in some manner to its algebraic structure.