Vector space

Linear space, over a field $K$

An Abelian group $E$, written additively, in which a multiplication of the elements by scalars is defined, i.e. a mapping \begin{equation} K\times E\rightarrow E\colon (\lambda,x)\rightarrow \lambda x, \end{equation} which satisfies the following axioms ($x,y\in E$; $\lambda,\mu,1\in K$):

1. $\lambda (x+y) = \lambda x + \lambda y$;
2. $(\lambda+\mu)x = \lambda x + \mu x$;
3. $(\lambda\mu)x=\lambda(\mu x)$;
4. $1x=x$.

Axioms 1.–4. imply the following important properties of a vector space ($0\in E$):

5. $\lambda 0=0$;
6. $0x=0$;
7. $(-1)x=-x$.

The elements of the vector space are called its points, or vectors; the elements of $K$ are called scalars.

The vector spaces most often employed in mathematics and in its applications are those over the field $C$ of complex numbers and over the field $R$ 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 $n$-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 $E$ is a subset $F\subset E$ 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 $x$ and $y$ of a vector space $E$ is the set of elements $z\in E$ of the form $z=\lambda x + (1-\lambda)y$, $\lambda\in K$. A set $G\in E$ 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: $G=x+F$; this means that each element $z\in G$ can be uniquely represented in the form $z=x+y$, $y\in F$, and that this equation realizes a one-to-one correspondence between $F$ and $G$.

The totality of all shifts $F_x=x+F$ of a given subspace $F$ forms a vector space over $K$, called the quotient space $E/F$, if the operations are defined as follows: \begin{equation} F_x+F_y=F_{x+y};\quad\lambda F_x=F_{\lambda x},\quad\lambda\in K. \end{equation}

Let $M=\{x_\alpha\}_{\alpha\in A}$ be an arbitrary set of vectors in $E$. A linear combination of the vectors $x_\alpha\in E$ is a vector $x$ defined by an expression \begin{equation} x=\sum_{\alpha}\lambda_\alpha x_\alpha,\quad\lambda_\alpha\in K, \end{equation} in which only a finite number of coefficients differ from zero. The set of all linear combinations of vectors of the set $M$ is the smallest subspace containing $M$ and is said to be the linear envelope of the set $M$. A linear combination is said to be trivial if all coefficients $\lambda_\alpha$ are zero. The set $M$ is said to be a linearly independent set if all non-trivial linear combinations of vectors in $M$ are non-zero.

Any linearly independent set is contained in some maximal linearly independent set $M_0$, i.e. in a set which ceases to be linearly independent after any element in $E$ has been added to it.

Each element $x\in E$ may be uniquely represented as a linear combination of elements of a maximal linearly independent set: \begin{equation} x=\sum_{\alpha}\lambda_\alpha x_\alpha,\quad x_\alpha\in M_0. \end{equation}

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 $K$ 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 $n$ elements is known as an $n$-dimensional space.

The theory of convex sets plays an important part in the theory of real and complex vector spaces. A set $M$ in a real vector space is said to be a convex set if for any two points $x$, $y$ in it the segment $tx + (1-t)y$, $t\in [0,1]$, also belongs to $M$.

The theory of linear functionals on vector spaces and the related theory of duality are important parts of the theory of vector spaces. Let $E$ be a vector space over a field $K$. An additive and homogeneous mapping $f\colon E\rightarrow K$, i.e. \begin{equation} f(x+y)=f(x)+f(y),\quad f(\lambda x)=\lambda f(x), \end{equation} is said to be a linear functional on $E$. The set $E^*$ of all linear functionals on $E$ forms a vector space over $K$ with respect to the operations \begin{equation} (f_1+f_2)(x)=f_1(x)+f_2(x),\quad (\lambda f)(x)=\lambda f(x),\quad x\in E,\quad\lambda\in K,\quad f_1,f_2,f\in E^*. \end{equation}

This vector space is said to be the conjugate, or dual, space of $E$. Several geometrical notions are connected with the concept of a conjugate space. Let $D\subset E$ (respectively, $\Gamma\subset E^*$); the set \begin{equation} D^\perp=\left\{f\in E^*\colon f(x)=0\quad \text{for all}\; x\in D\right\}, \end{equation} or \begin{equation} \Gamma_\perp=\left\{x\in E\colon f(x)=0\quad \text{for all}\; f\in\Gamma\right\}, \end{equation} is said to be the annihilator or orthogonal complement of $D$ (respectively, of $\Gamma$); here $D^\perp$ and $\Gamma_\perp$ are subspaces of $E^*$ and $E$, respectively. If $f$ is a non-zero element of $E^*$, $\{ f\}_\perp$ is a maximal proper linear subspace in $E$, which is sometimes called a hypersubspace; a shift of such a subspace is said to be a hyperplane in $E$; thus, any hyperplane has the form \begin{equation} \{x\colon f(x)=\lambda\},\quad f\neq 0,\quad f\in E^*,\quad\lambda\in K. \end{equation}

If $F$ is a subspace of the vector space $E$, there exist natural isomorphisms between $F^*$ and $E^*/F^\perp$ and between $(E/F)^*$ and $F^\perp$.

A subset $\Gamma\subset E^*$ is said to be a total subset over $E$ if its annihilator contains only the zero element, $\Gamma_\perp=\{ 0\}$.

Each linearly independent set $\{ x_\alpha\}_{\alpha\in A}\subset E$ can be brought into correspondence with a conjugate set $\{ f_\alpha\}_{\alpha\in A}\subset E^*$, i.e. with a set such that $f_\alpha(x_\beta)=\delta_{\alpha\beta}$ (the Kronecker symbol) for all $\alpha$, $\beta\in A$. The set of pairs $\{ x_\alpha,f_\alpha\}$ is said to be a biorthogonal system. If the set $\{ x_\alpha\}$ is a basis in $E$, then $\{ f_\alpha\}$ is total over $E$.

An important chapter in the theory of vector spaces is the theory of linear transformations of these spaces. Let $E_1$, $E_2$ be two vector spaces over the same field $K$. Then an additive and homogeneous mapping $T$ of $E_1$ into $E_2$, i.e. \begin{equation} T(x+y)=Tx+Ty;\quad T(\lambda x)=\lambda Tx;\quad x,y\in E_1, \end{equation} is said to be a linear mapping or linear operator, mapping $E_1$ into $E_2$ (or from $E_1$ into $E_2$). A special case of this concept is a linear functional, or a linear operator from $E_1$ into $K$. An example of a linear mapping is the natural mapping from $E$ into the quotient space $E/F$, which establishes a one-to-one correspondence between each element $x\in E$ and the flat set $F_x\in E/F$. The set $\mathcal{L}(E_1,E_2)$ of all linear operators $T\colon E_1\rightarrow E_2$ forms a vector space with respect to the operations \begin{equation} (T_1+T_2)x=T_1x+T_2y;\quad (\lambda T)x=\lambda Tx;\quad x\in E_1;\quad\lambda\in K;\quad T_1,T_2,T\in\mathcal{L}(E_1,E_2). \end{equation}

Two vector spaces $E_1$ and $E_2$ are said to be isomorphic if there exists a linear operator (an isomorphism) which realizes a one-to-one correspondence between their elements. $E_1$ and $E_2$ are isomorphic if and only if their bases have equal cardinalities.

Let $T$ be a linear operator from $E_1$ into $E_2$. The conjugate linear operator, or dual linear operator, of $T$ is the linear operator $T^*$ from $E_1^*$ into $E_2^*$ defined by the equation \begin{equation} (T^*\phi)(X)=\phi(Tx)\quad\text{for all}\;x\in E_1,\phi\in E_2^*. \end{equation}

The relations $T^{*^{-1}}(0)=[T(E_1)]^{\perp}$, $T^*(E_2^*)=[T^{-1}(0)]^{\perp}$ are valid, which imply that $T^*$ is an isomorphism if and only if $T$ 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.

Problems of extending linear mappings are an important group of problems in the theory of vector spaces. Let $F$ be a subspace of a vector space $E_1$, let $E_2$ be a linear space over the same field as $E_1$ and let $T_0$ be a linear mapping from $F$ into $E_2$; it is required to find an extension $T$ of $T_0$ which is defined on all of $E_1$ and which is a linear mapping from $E_1$ into $E_2$. 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 $\{E_\alpha\}_{\alpha\in I}$ be a family of vector spaces over a field $K$. The set $E$ which is the product of $E_\alpha$ can be made into a vector space over $K$ by introducing the operations \begin{equation} (x_\alpha)+(y_\alpha)=(x_\alpha+y_\alpha);\quad\lambda(x_\alpha)=(\lambda x_\alpha);\quad \lambda\in K;\quad x_\alpha,y_\alpha\in E_\alpha,\quad \alpha\in I. \end{equation}

The resulting vector space $E$ is called the direct product of the vector spaces $E_\alpha$, and is written as $\prod_{\alpha\in I}E_\alpha$. The subspace of the vector space $E$ consisting of all sequences $(x_\alpha)$ for each of which the set $\{\alpha\colon x_\alpha\neq 0\}$ is finite, is said to be the direct sum of the vector spaces $E_\alpha$, and is written as $\sum_{\alpha}E_\alpha$ or $\oplus_{\alpha}E_\alpha$. These two notions coincide if the number of terms is finite. In this case one uses the notations: \begin{equation} E_1+\ldots+E_n,\quad E_1\oplus\ldots\oplus E_n\quad\text{or}\quad E_1\times\ldots\times E_n. \end{equation}

Let $E_1$ and $E_2$ be vector spaces over the same field $K$; let $E_1'$, $E_2'$ be total subspaces of the vector spaces $E_1^*$, $E_2^*$, and let $E_1\Box E_2$ be the vector space with the set of all elements of the space $E_1\times E_2$ as its basis. Each element $x\Box y\in E_1\Box E_2$ can be brought into correspondence with a bilinear function $b=T(x,y)$ on $E_1'\times E_2'$ using the formula $b(f,g)=f(x)g(y)$, $f\in E_1'$, $g\in E_2'$. This mapping on the basis vectors $x\Box y\in E_1\Box E_2$ may be extended to a linear mapping $T$ from the vector space $E_1\Box E_2$ into the vector space of all bilinear functionals on $E_1'\times E_2'$. Let $E_0=T^{-1}(0)$. The tensor product of $E_1$ and $E_2$ is the quotient space $E_1\otimes E_2=(E_1\Box E_2)/E_0$; the image of the element $x\Box y$ is written as $x\otimes y$. The vector space $E_1\otimes E_2$ is isomorphic to the vector space of bilinear functionals on $E_1'\times E_2'$ (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.

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