البوابة
من طرف Vector space,
linear space, over a field
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
linear space, over a field
An Abelian group
, written additively, in which a multiplication of the elements by scalars is defined, i.e. a mapping  |
; ):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
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.
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 
.
of a given subspace 
forms a vector space over 
, called the quotient space 
, if the operations are defined as follows: 
be an arbitrary set of vectors in 
. A linear combination of the vectors 
is a vector 
defined by an expression 
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.
, i.e. in a set which ceases to be linearly independent after any element in 
has been added to it.
may be uniquely represented as a linear combination of elements of a maximal linearly independent set: 
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.
in a real vector space is said to be a convex set if for any two points 
in it the segment 
, 
, also belongs to 
.
be a vector space over a field 
. An additive and homogeneous mapping 
, i.e. 
. The set 
of all linear functionals on 
forms a vector space over 
with respect to the operations 
. Several geometrical notions are connected with the concept of a conjugate space. Let 
(respectively, 
); the set 
, 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 
is a subspace of the vector space 
, there exist natural isomorphisms between 
and 
and between 
and 
.
is said to be a total subset over 
if its annihilator contains only the zero element, 
.
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 
.
be two vector spaces over the same field 
. Then an additive and homogeneous mapping 
of 
into 
, i.e. 
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 
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.
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 
, 
are valid, which imply that 
is an isomorphism if and only if 
is an isomorphism.
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.
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 
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: 
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).
